SELINA Solutions for Class 10 Maths Chapter 15 - Similarity (With Applications to Maps and Models)

Chapter 15 - Similarity (With Applications to Maps and Models) Exercise Ex. 15(A)

Question 1

In the figure, given below, straight lines AB and CD intersect at P; and AC BD. Prove that :

If BD = 2.4 cm, AC = 3.6 cm, PD = 4.0 cm and PB = 3.2 cm; find the lengths of PA and PC.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 1

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 2

In the figure, given below, straight lines AB and CD intersect at P; and AC BD. Prove that:

ΔAPC and ΔBPD are similar.

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 2

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 3

In a trapezium ABCD, side AB is parallel to side DC; and the diagonals AC and BD intersect each other at point P. Prove that :

Δ APB is similar to Δ CPD.

Solution 3

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 4

In a trapezium ABCD, side AB is parallel to side DC; and the diagonals AC and BD intersect each other at point P. Prove that :

PA x PD = PB x PC. 

Solution 4

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 5

P is a point on side BC of a parallelogram ABCD. If DP produced meets AB produced at point L, prove that :

DP : PL = DC : BL.

Solution 5

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 6

P is a point on side BC of a parallelogram ABCD. If DP produced meets AB produced at point L, prove that :

DL : DP = AL : DC.

Solution 6

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 7

In quadrilateral ABCD, the diagonals AC and BD intersect each other at point O. If AO = 2CO and BO = 2DO; show that :

Δ AOB is similar to Δ COD.

Solution 7

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 8

In quadrilateral ABCD, the diagonals AC and BD intersect each other at point O. If AO = 2CO and BO = 2DO; show that :

OA x OD = OB x OC.

Solution 8

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

 

 

 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 9

In Δ ABC, angle ABC is equal to twice the angle ACB, and bisector of angle ABC meets the opposite side at point P. Show that :

CB : BA = CP : PA

Solution 9

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 10

In Δ ABC, angle ABC is equal to twice the angle ACB, and bisector of angle ABC meets the opposite side at point P. Show that :

AB x BC = BP x CA 

Solution 10

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 11

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 11

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 12

In the given figure, DE BC, AE = 15 cm, EC = 9 cm, NC = 6 cm and BN = 24 cm.

Write all possible pairs of similar triangles.

 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 12

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 13

In the given figure, DE BC, AE = 15 cm, EC = 9 cm, NC = 6 cm and BN = 24 cm.

Find lengths of ME and DM.

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 13

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 14

In the given figure, AD = AE and AD2 = BD x EC.

Prove that: triangles ABD and CAE are similar. 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 14

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 15

In the given figure, AB DC, BO = 6 cm and DQ = 8 cm; find: BP x DO.

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 15

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 16

Angle BAC of triangle ABC is obtuse and AB = AC. P is a point in BC such that PC = 12 cm. PQ and PR are perpendiculars to sides AB and AC respectively. If PQ = 15 cm and PR = 9 cm; find the length of PB.

Solution 16

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 17

State, true or false:

(i) Two similar polygons are necessarily congruent.

(ii) Two congruent polygons are necessarily similar.

(iii) All equiangular triangles are similar.

(iv) All isosceles triangles are similar.

(v) Two isosceles-right triangles are similar.

(vi) Two isosceles triangles are similar, if an angle of one is congruent to the corresponding angle of the other.

(vii) The diagonals of a trapezium, divide each other into proportional segments.

Solution 17

(i) False

(ii) True

(iii) True

(iv) False

(v) True

(vi) True

(vii) True

Question 18

Given: GHE = DFE = 90o, DH = 8, DF = 12, DG = 3x - 1 and DE = 4x + 2.

Find: the lengths of segments DG and DE.

Solution 18

Question 19

D is a point on the side BC of triangle ABC such that angle ADC is equal to angle BAC. Prove that: CA2 = CB CD.

Solution 19

Question 20

In the given figure, ∆ABC and ∆AMP are right angled at B and M respectively.

Given AC = 10 cm, AP = 15 cm and PM = 12 cm.

  1. ∆ ABC ~ ∆ AMP.
  2. Find AB and BC.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 20

(i)  In ∆ ABC and ∆ AMP,

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsBAC= Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsPAM [Common]

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsABC= Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsPMA [Each = 90°]

∆ ABC ~ ∆ AMP [AA Similarity]

(ii)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

 

 

 

 

 

 

 

 

Question 21

Given: RS and PT are altitudes of PQR. Prove that:

(i)

(ii) PQ QS = RQ QT.

Solution 21

(i)

(ii)

Since, triangles PQT and RQS are similar.

Question 22

Given: ABCD is a rhombus, DPR and CBR are straight lines. Prove that: DP Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models CR = DC Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models PR.

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 22

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Hence, DP Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models CR = DC Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models PR

Question 23

Given: FB = FD, AE FD and FC AD. Prove:

Solution 23

Question 24

In PQR, Q = 90o and QM is perpendicular to PR. Prove that:

(i) PQ2 = PM PR

(ii) QR2 = PR MR

(iii) PQ2 + QR2 = PR2

Solution 24

(i) In PQM and PQR,

PMQ = PQR = 90o

QPM = RPQ (Common)

(ii) In QMR and PQR,

QMR = PQR = 90o

QRM = QRP (Common)

(iii) Adding the relations obtained in (i) and (ii), we get,

Question 25

In Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsABC, Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsB = 90o and BD Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models AC.

(i) If CD = 10 cm and BD = 8 cm; find AD.

(ii) If AC = 18 cm and AD = 6 cm; find BD.

(iii) If AC = 9 cm and AB = 7 cm; find AD.

Solution 25

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(i) In Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsCDB,

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models1 + Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models2 +Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models3 = 180o

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models1 + Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models3 = 90o ..... (1)(Since, Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models2 = 90o)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models3 + Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models4 = 90o .....(2) (Since, Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsABC = 90o)

From (1) and (2),

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models1 + Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models3 = Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models3 + Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models4

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models1 = Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models4

Also, Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models2 = Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models5 = 90o

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Hence, AD = 6.4 cm

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

 

(iii)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 26

In the figure, PQRS is a parallelogram with PQ = 16 cm and QR = 10 cm. L is a point on PR such that RL: LP = 2: 3. QL produced meets RS at M and PS produced at N.

Find the lengths of PN and RM.

Solution 26

Question 27

In quadrilateral ABCD, diagonals AC and BD intersect at point E such that AE: EC = BE: ED. Show that ABCD is a trapezium.

Solution 27

Given, AE: EC = BE: ED

Draw EF || AB

In ABD, EF || AB

Using Basic Proportionality theorem,

Thus, in DCA, E and F are points on CA and DA respectively such that

Thus, by converse of Basic proportionality theorem, FE || DC.

But, FE || AB.

Hence, AB || DC.

Thus, ABCD is a trapezium.

Question 28

In triangle ABC, AD is perpendicular to side BC and AD2 = BD DC. Show that angle BAC = 90o.

Solution 28

Given, AD2 = BD DC

So, these two triangles will be equiangular.

Question 29

In the given figure, AB || EF || DC; AB = 67.5 cm, DC = 40.5 cm and AE = 52.5 cm.

(i) Name the three pairs of similar triangles.

(ii) Find the lengths of EC and EF.

Solution 29

(i) The three pair of similar triangles are:

BEF and BDC

CEF and CAB

ABE and CDE

(ii) Since, ABE and CDE are similar,

Since, CEF and CAB are similar,

Question 30

In the given figure, QR is parallel to AB and DR is parallel to QB. Prove that: PQ2 = PD PA.

Solution 30

Given, QR is parallel to AB. Using Basic proportionality theorem,

Also, DR is parallel to QB. Using Basic proportionality theorem,

From (1) and (2), we get,

Question 31

Through the mid-point M of the side CD of a parallelogram ABCD, the line BM is drawn intersecting diagonal AC in L and AD produced in E. Prove that: EL = 2BL.

Solution 31

1 = 6 (Alternate interior angles)

2 = 3 (Vertically opposite angles)

DM = MC (M is the mid-point of CD)

So, DE = BC (Corresponding parts of congruent triangles)

Also, AD = BC (Opposite sides of a parallelogram)

AE = AD + DE = 2BC

Now, 1 = 6 and 4 = 5

Question 32

In the figure, given below, P is a point on AB such that AP: PB = 4: 3. PQ is parallel to AC.

(i) Calculate the ratio PQ: AC, giving reason for your answer.

(ii) In triangle ARC, ARC = 90o and in triangle PQS, PSQ = 90o. Given QS = 6 cm, calculate the length of AR.

Solution 32

(i) Given, AP: PB = 4: 3.

Since, PQ || AC. Using Basic Proportionality theorem,

Now, PQB = ACB (Corresponding angles)

QPB = CAB (Corresponding angles)

(ii) ARC = QSP = 90o

ACR = SPQ (Alternate angles)

Question 33

In the right-angled triangle QPR, PM is an altitude. Given that QR = 8 cm and MQ = 3.5 cm, calculate the value of PR.

Solution 33

We have:

Question 34

In the figure, given below, the medians BD and CE of a triangle ABC meet at G. Prove that:

(i) and

(ii) BG = 2 GD from (i) above.

Solution 34

(i) Since, BD and CE are medians.

AD = DC

AE = BE

Hence, by converse of Basic Proportionality theorem,

ED || BC

In EGD and CGB,

(ii) Since,

In AED and ABC,

From (1),

Chapter 15 - Similarity (With Applications to Maps and Models) Exercise Ex. 15(B)

Question 1

In the following figure, point D divides AB in the ratio 3 : 5. Find :

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 1

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Now, DE is parallel to BC.

Then, by Basic proportionality theorem, we have

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsSelina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 2

In the following figure, point D divides AB in the ratio 3 : 5. Find :

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 2

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 3

In the following figure, point D divides AB in the ratio 3 : 5. Find :

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 3

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 4

In the following figure, point D divides AB in the ratio 3 : 5. Find :

DE = 2.4 cm, find the length of BC.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 4

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 5

In the following figure, point D divides AB in the ratio 3 : 5. Find :

BC = 4.8 cm, find the length of DE.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 5

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 6

In the given figure, PQ AB; CQ = 4.8 cm QB = 3.6 cm and AB = 6.3 cm. Find :

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 6

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 7

In the given figure, PQ AB; CQ = 4.8 cm QB = 3.6 cm and AB = 6.3 cm. Find :

PQ

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 7

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 8

In the given figure, PQ AB; CQ = 4.8 cm QB = 3.6 cm and AB = 6.3 cm. Find :

If AP = x, then the value of AC in terms of x. Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 8

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 9

A line PQ is drawn parallel to the side BC of Δ ABC which cuts side AB at P and side AC at Q. If AB = 9.0 cm, CA = 6.0 cm and AQ = 4.2 cm, find the length of AP.

Solution 9

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 10

In Δ ABC, D and E are the points on sides AB and AC respectively.

Find whether DE BC, if

AB = 9cm, AD = 4cm, AE = 6cm and EC = 7.5cm. 

Solution 10

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 11

In Δ ABC, D and E are the points on sides AB and AC respectively.

Find whether DE BC, if AB = 6.3 cm, EC = 11.0 cm, AD =0.8 cm and EA = 1.6 cm.

Solution 11

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

 

 

 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

Question 12

In the given figure, Δ ABC ~ Δ ADE. If AE: EC = 4 : 7 and DE = 6.6 cm, find BC. If 'x' be the length of the perpendicular from

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

A to DE, find the length of perpendicular from A to BC in terms of 'x'.

Solution 12

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

 Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 13

A line segment DE is drawn parallel to base BC of Δ ABC which cuts AB at point D and AC at point E. If AB = 5BD and EC = 3.2 cm, find the length of AE.

Solution 13

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

 

 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 14

In the figure, given below, AB, CD and EF are parallel lines. Given AB = 7.5 cm, DC = y cm, EF = 4.5 cm, BC = x cm and CE = 3. cm, calculate the values of x and y.

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

Solution 14

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

 

Question 15

In the figure, given below, PQR is a right-angle triangle right angled at Q. XY is parallel to QR, PQ = 6 cm, PY = 4 cm and PX : XQ = 1 : 2. Calculate the lengths of PR and QR.

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 15

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 16

In the following figure, M is mid-point of BC of a parallelogram ABCD. DM intersects the diagonal AC at P and AB produced at E.

Prove that : PE = 2 PD 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 16

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 17

The given figure shows a parallelogram ABCD. E is a point in AD and CE produced meets BA produced at point F. If AE = 4 cm, AF = 8 cm and AB = 12 cm, find the perimeter of the parallelogram ABCD.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 17

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Chapter 15 - Similarity (With Applications to Maps and Models) Exercise Ex. 15(C)

Question 1

(i) The ratio between the corresponding sides of two similar triangles is 2 is to 5. Find the ratio between the areas of these triangles.

(ii) Areas of two similar triangles are 98 sq. cm and 128 sq. cm. Find the ratio between the lengths of their corresponding sides.

Solution 1

We know that the ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding sides.

 

(i) Required ratio = Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(ii) Required ratio = Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 2

A line PQ is drawn parallel to the base BC of Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsABC which meets sides AB and AC at points P and Q respectively. If AP =Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsPB; find the value of:

(i) Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(ii) Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 2

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

 

(i) AP =Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsPB Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

 

(ii) Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 3

The perimeters of two similar triangles are 30 cm and 24 cm. If one side of the first triangle is 12 cm, determine the corresponding side of the second triangle.

Solution 3

Let Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 4

In the given figure, AX: XB = 3: 5.

Find:

(i) the length of BC, if the length of XY is 18 cm.

(ii) the ratio between the areas of trapezium XBCY and triangle ABC.

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 4

Given, Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(i)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(ii)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 5

ABC is a triangle. PQ is a line segment intersecting AB in P and AC in Q such that PQ || BC and divides triangle ABC into two parts equal in area. Find the value of ratio BP: AB.

Solution 5

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

From the given information, we have:

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 6

In the given triangle PQR, LM is parallel to QR and PM: MR = 3: 4.

Calculate the value of ratio:

(i) Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(ii) Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(iii) Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 6

(i)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(ii) Since Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsLMN and Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsMNR have common vertex at M and their bases LN and NR are along the same straight line

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

 

(iii) Since Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsLQM and Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsLQN have common vertex at L and their bases QM and QN are along the same straight line

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 7

The given diagram shows two isosceles triangles which are similar also. In the given diagram, PQ and BC are not parallel; PC = 4, AQ = 3, QB = 12, BC = 15 and AP = PQ. Calculate:

(i) the length of AP,

(ii) the ratio of the areas of triangle APQ and triangle ABC.

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 7

(i)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(ii)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 8

In the figure, given below, ABCD is a parallelogram. P is a point on BC such that BP: PC = 1: 2. DP produced meets AB produces at Q. Given the area of triangle CPQ = 20 cm2. Calculate:

(i) area of triangle CDP,

(ii) area of parallelogram ABCD.

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 8

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 9

In the given figure, BC is parallel to DE. Area of triangle ABC = 25 cm2, Area of trapezium BCED = 24 cm2 and DE = 14 cm. Calculate the length of BC. Also, find the area of triangle BCD.

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 9

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 10

The given figure shows a trapezium in which AB is parallel to DC and diagonals AC and BD intersects at point P. If AP: CP = 3: 5, find:

(i) Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsAPB : Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsCPB(ii) Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsDPC : Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsAPB

(iii) Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsADP : Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsAPB(iv) Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsAPB : Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsADB

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 10

(i) Since Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsAPB and Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsCPB have common vertex at B and their bases AP and PC are along the same straight line

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(ii) Since Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsDPC and Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsBPA are similar

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(iii) Since Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsADP and Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsAPB have common vertex at A and their bases DP and PB are along the same straight line

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(iv) Since Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsAPB and Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsADB have common vertex at A and their bases BP and BD are along the same straight line

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 11

In the given figure, ABC is a triangle. DE is parallel to BC and Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(i) Determine the ratios Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(ii) Prove that Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsDEF is similar to Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsCBF. Hence, find Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models.

(iii) What is the ratio of the areas of Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsDEF and Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsBFC?

Solution 11

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

 

(i) Given, DE || BC and Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

In Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsADE and Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsABC,

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsA = Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsA(Corresponding Angles)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsADE = Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsABC(Corresponding Angles)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsSelina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models(By AA- similarity)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsSelina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models..........(1)

NowSelina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Using (1), we getSelina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models.........(2)

(ii) In Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsDEF and Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsCBF,

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsFDE = Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsFCB(Alternate Angle)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsDFE = Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsBFC(Vertically Opposite Angle)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsSelina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsDEF Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsSelina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsCBF(By AA- similarity)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Modelsusing (2)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models.

(iii) Since the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides, therefore

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 12

In the given figure, Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsB = Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsE, Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsACD = Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsBCE, AB = 10.4 cm and DE = 7.8 cm. Find the ratio between areas of the Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsABC and Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsDEC.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 12

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Chapter 15 - Similarity (With Applications to Maps and Models) Exercise Ex. 15(D)

Question 1

A triangle ABC has been enlarged by scale factor m = 2.5 to the triangle A' B' C' Calculate :

the length of AB, if A' B' = 6 cm. 

Solution 1

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 2

A triangle ABC has been enlarged by scale factor m = 2.5 to the triangle A' B' C' Calculate :

the length of C' A' if CA = 4 cm. 

Solution 2

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 3

A triangle LMN has been reduced by scale factor 0.8 to the triangle L' M' N'. Calculate:

the length of M' N', if MN = 8 cm.

Solution 3

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 4

A triangle LMN has been reduced by scale factor 0.8 to the triangle L' M' N'. Calculate:

the length of LM, if L' M' = 5.4 cm.

Solution 4

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 5

A triangle ABC is enlarged, about the point 0 as centre of enlargement, and the scale factor is 3. Find : OC', if OC = 21 cm

Also, state the value of : 

(a) Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

(b) Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 5

Given that triangle ABC is enlarged and the scale factor is m = 3 to the triangle A'B'C'.

OC = 21 cm

So, (OC)3 = OC'

i.e. 21 x 3 = OC'

i.e. OC' = 63 cm


Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 6

A triangle ABC is enlarged, about the point O as centre of enlargement, and the scale factor is 3. Find :

A' B', if AB = 4 cm.

Solution 6

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 7

A triangle ABC is enlarged, about the point 0 as centre of enlargement, and the scale factor is 3. Find :

BC, if B' C' = 15 cm. 

Solution 7

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 8

A triangle ABC is enlarged, about the point 0 as centre of enlargement, and the scale factor is 3. Find :

OA, if OA' = 6 cm. 

Solution 8

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 9

A model of an aeroplane is made to a scale of 1 : 400. Calculate :

the length, in cm, of the model; if the length of the aeroplane is 40 m. 

Solution 9

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 10

A model of an aeroplane is made to a scale of 1 : 400. Calculate :

the length, in m, of the aeroplane, if length of its model is 16 cm.

Solution 10

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 11

The dimensions of the model of a multistorey building are 1.2 m x 75 cm x 2 m. If the scale factor is 1 : 30; find the actual dimensions of the building.

Solution 11

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 12

On a map drawn to a scale of 1 : 2,50,000; a triangular plot of land has the following measurements : AB = 3 cm, BC = 4 cm and  ABC = 90°.

Calculate :

the actual lengths of AB and BC in km. 

Solution 12

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 13

On a map drawn to a scale of 1 : 2,50,000; a triangular plot of land has the following measurements : AB = 3 cm, BC = 4 cm and angle ABC = 90°.

Calculate :

the area of the plot in sq. km.

Solution 13

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 14

A model of a ship is made to a scale 1 : 300.

 i. The length of the model of the ship is 2 m. Calculate the length of the ship.

 ii. The area of the deck of the ship is 180,000 m2. Calculate the area of the deck of the model.

 iii. The volume of the model is 6.5 m3. Calculate the volume of the ship. 

Solution 14

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Chapter 15 - Similarity (With Applications to Maps and Models) Exercise Ex. 15(E)

Question 1

In the following figure, XY is parallel to BC, AX = 9 cm, XB = 4.5 cm and BC = 18 cm.

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 1

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 2

In the following figure, XY is parallel to BC, AX = 9 cm, XB = 4.5 cm and BC = 18 cm.

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 2

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 3

In the following figure, ABCD to a trapezium with AB DC. If AB = 9 cm, DC = 18 cm, CF= 13.5,cm, AP = 6 cm and BE = 15 cm, Calculate: PE

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 3

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 4

In the following figure, ABCD to a trapezium with AB DC. If AB = 9 cm, DC = 18 cm, CF= 13.5,cm, AP = 6 cm and BE = 15 cm, Calculate: CE

 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

Solution 4

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 5

In the following figure, ABCD to a trapezium with AB DC. If AB = 9 cm, DC = 18 cm, CF= 13.5,cm, AP = 6 cm and BE = 15 cm, Calculate: AF

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 5

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 6

In the following figure, AB, CD and EF are perpendicular to the straight line BDF.

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 6

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 7

Triangle ABC is similar to triangle PQR. If AD and PM are corresponding medians of the two triangles, prove that: Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models.

Solution 7

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 8

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 8

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 9

 

Triangle ABC is similar to triangle PQR. If bisector of angle BAC meets BC at point D and bisector of angle QPR meets QR at point M, prove that:Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models.

Solution 9

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 10

In the following figure, AXY = AYX. If Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Modelsshow that triangle ABC is isosceles.

 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 10

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 11

In the following diagram, lines l, m and n are parallel to each other. Two transversals p and q intersect the parallel lines at points A, B, C and P, Q, R as shown. 

Prove that: Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 11

Join AR.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

In Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsACR, BX || CR. By Basic Proportionality theorem,

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

In Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsAPR, XQ || AP. By Basic Proportionality theorem,

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

From (1) and (2), we get,

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 12

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 12

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 13

In the figure given below, AB EF CD. If AB = 22.5 cm, EP = 7.5 cm, PC = 15 cm and DC = 27 cm. Calculate : EF

 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 13

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 14

In the figure given below, AB EF CD. If AB = 22.5 cm, EP = 7.5 cm, PC = 15 cm and DC = 27 cm. Calculate : AC

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 14

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 15

In ΔABC, ABC = DAC, AB = 8 cm, AC = 4 cm and AD = 5 cm.

Prove that ΔACD is similar to ΔBCA.

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 15

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 16

In ΔABC, ABC = DAC, AB = 8 cm, AC = 4 cm and AD = 5 cm.

Find BC and CD.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 16

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 17

In ΔABC, ABC = DAC, AB = 8 cm, AC = 4 cm and AD = 5 cm.

Find area of ΔACD : area of ΔABC.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 17

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 18

In the given triangle P, Q and R are mid-points of sides AB, BC and AC respectively. Prove that triangle QRP is similar to triangle ABC.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 18

In Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsABC, PR || BC. By Basic proportionality theorem,

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Also, in Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsPAR and Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsABC,

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Similarly, Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 19

In the following figure, AD and CE are medians of Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsABC. DF is drawn parallel to CE. Prove that:

(i) EF = FB,

(ii) AG: GD = 2: 1

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 19

(i)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(ii) In Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsAFD, EG || FD. Using Basic Proportionality theorem,

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models … (1)

Now, AE = EB (as E is the mid-point of AB)

AE = 2EF (Since, EF = FB, by (i))

From (1),

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Hence, AG: GD = 2: 1.

Question 20

Two similar triangles are equal in area. Prove that the triangles are congruent.

Solution 20

Let us assume two similar triangles as Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsABC Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsPQR

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 21

The ratio between the altitudes of two similar triangles is 3: 5; write the ratio between their:

(i) medians. (ii) perimeters. (iii) areas.

Solution 21

The ratio between the altitudes of two similar triangles is same as the ratio between their sides.

(i) The ratio between the medians of two similar triangles is same as the ratio between their sides.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsRequired ratio = 3: 5

(ii) The ratio between the perimeters of two similar triangles is same as the ratio between their sides.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsRequired ratio = 3: 5

(iii) The ratio between the areas of two similar triangles is same as the square of the ratio between their corresponding sides.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsRequired ratio = (3)2 : (5)2 = 9: 25

Question 22

The ratio between the areas of two similar triangles is 16: 25. Find the ratio between their:

(i) perimeters. (ii) altitudes. (iii) medians.

Solution 22

The ratio between the areas of two similar triangles is same as the square of the ratio between their corresponding sides.

So, the ratio between the sides of the two triangles = 4: 5

(i) The ratio between the perimeters of two similar triangles is same as the ratio between their sides.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsRequired ratio = 4: 5

(ii) The ratio between the altitudes of two similar triangles is same as the ratio between their sides.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsRequired ratio = 4: 5

(iii) The ratio between the medians of two similar triangles is same as the ratio between their sides.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsRequired ratio = 4: 5

Question 23

The following figure shows a triangle PQR in which XY is parallel to QR. If PX: XQ = 1: 3 and QR = 9 cm, find the length of XY.

Further, if the area of Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsPXY = x cm2; find, in terms of x, the area of:

(i) triangle PQR. (ii) trapezium XQRY.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 23

In Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsPXY and Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsPQR, XY is parallel to QR, so corresponding angles are equal.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Hence, Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models (By AA similarity criterion)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

 (i) We know that the ratio of areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

 (ii) Ar (trapezium XQRY) = Ar (Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsPQR) - Ar (Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsPXY)

= (16x - x) cm2

= 15x cm2

Question 24

On a map drawn to a scale of 1: 20000, a rectangular plot of land ABCD is measured as AB = 24 cm and BC = 32 cm. Calculate:

(i) the diagonal distance of the plot in kilometre.

(ii) the area of the plot in sq. km.

Solution 24

Scale :- 1 : 20000

1 cm represents 20000 cm= Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models = 0.2 km

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(i)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

= Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

= 576 + 1024 = 1600

AC = 40 cm

 

Actual length of diagonal = 40 Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 0.2 km = 8 km

(ii)

1 cm represents 0.2 km

1 cm2 represents 0.2 Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 0.2 Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

The area of the rectangle ABCD = AB Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models BC

= 24 Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 32 = 768Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Actual area of the plot = 0.2 Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 0.2 Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 768 Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models = 30.72 km2

Question 25

The dimensions of the model of a multistoreyed building are 1 m by 60 cm by 1.20 m. If the scale factor is 1: 50, find the actual dimensions of the building. Also, find:

(i) the floor area of a room of the building, if the floor area of the corresponding room in the model is 50 sq. cm.

(ii) the space (volume) inside a room of the model, if the space inside the corresponding room of the building is 90 m3.

Solution 25

The dimensions of the building are calculated as below.

Length = 1 Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 50 m = 50 m

Breadth = 0.60 Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 50 m = 30 m

Height = 1.20 Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 50 m = 60 m

Thus, the actual dimensions of the building are 50 m Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 30 m Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 60 m.

(i)

Floor area of the room of the building Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(ii)

Volume of the model of the building

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 26

In a triangle PQR, L and M are two points on the base QR, such that Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsLPQ = Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsQRP and Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsRPM = Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsRQP. Prove that:

(i) Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(ii) Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(iii) Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 26

(i)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(ii)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(iii)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 27

A triangle ABC with AB = 3 cm, BC = 6 cm and AC = 4 cm is enlarged to Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsDEF such that the longest side of Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsDEF = 9 cm. Find the scale factor and hence, the lengths of the other sides of Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsDEF.

Solution 27

Triangle ABC is enlarged to DEF. So, the two triangles will be similar.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Longest side in Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsABC = BC = 6 cm

Corresponding longest side in Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsDEF = EF = 9 cm

Scale factor = Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models= 1.5

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 28

Two isosceles triangles have equal vertical angles. Show that the triangles are similar.

If the ratio between the areas of these two triangles is 16: 25, find the ratio between their corresponding altitudes.

Solution 28

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Let ABC and PQR be two isosceles triangles.

Then, Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Also, Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsA = Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsP (Given)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Let AD and PS be the altitude in the respective triangles.

We know that the ratio of areas of two similar triangles is equal to the square of their corresponding altitudes.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 29

In Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsABC, AP: PB = 2: 3. PO is parallel to BC and is extended to Q so that CQ is parallel to BA. Find:

(i) area Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsAPO : area Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsABC.

(ii) area Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsAPO : area Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsCQO.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 29

In triangle ABC, PO || BC. Using Basic proportionality theorem,

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(i)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(ii)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 30

The following figure shows a triangle ABC in which AD and BE are perpendiculars to BC and AC respectively. Show that:

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 30

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 31

In the given figure, ABC is a triangle with Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsEDB = Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsACB. Prove that Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsABC Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsEBD. If BE = 6 cm, EC = 4 cm, BD = 5 cm and area of Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsBED = 9 cm2. Calculate the :

(i) length of AB.

(ii) area of Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsABC.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 31

In Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsABC and Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsEBD,

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsACB = Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsEDB (given)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsABC = Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsEBD (common)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models (by AA- similarity)

(i) We have, Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(ii) Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 32

In the given figure, ABC is a right angled triangle with mBAC = 90°

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

  1. Prove that ∆ADB ~ ∆CDA
  2. If BD = 18 cm and CD = 8 cm, find AD.
  3. Find the ratio of the area of ∆ADB to the area of ∆CDA.
Solution 32

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 33

In the given figure, AB and DE are perpendiculars to BC.

Prove that : ΔABC ~  ΔDEC 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 33

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 34

In the given figure, AB and DE are perpendiculars to BC.

If AB = 6 cm, DE = 4 cm and AC = 15 cm. Calculate CD.

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 34

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 35

In the given figure, AB and DE are perpendiculars to BC.

Find the ratio of the area of a ΔABC : area of ΔDEC.

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 35

 

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

 

 

Question 36

ABC is a right angled triangle with ABC = 90°. D is any point on AB and DE is perpendicular to AC. Prove that :

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

 

ΔADE ~ ΔACB.

Solution 36

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 37

ABC is a right angled triangle with ABC = 90°. D is any point on AB and DE is perpendicular to AC. Prove that :

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

 

If AC = 13 cm, BC = 5 cm and AE = 4 cm. Find DE and AD.

Solution 37

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Question 38

ABC is a right angled triangle with ABC = 90°. D is any point on AB and DE is perpendicular to AC. Prove that :

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

 

 

Find, area of ΔADE : area of quadrilateral BCED.

Solution 38

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 39

Given: AB || DE and BC || EF. Prove that:

(i) Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

(ii) Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

 

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Solution 39

(i) In Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsAGB, DE || AB , by Basic proportionality theorem,

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models .... (1)


 

In Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And ModelsGBC, EF || BC, by Basic proportionality theorem,

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models .... (2)


 

From (1) and (2), we get,

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models


 

(ii)

From (i), we have:

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Question 40

PQR is a triangle. S is a point on the side QR of ΔPQR such that PSR = QPR. Given QP = 8 cm, PR = 6 cm and SR = 3 cm.

i. Prove ΔPQR ∼ ΔSPR.

ii. Find the lengths of QR and PS.

iii. Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models 

Solution 40

i.

In PQR and SPR,

PSR = QPR … given

PRQ = PRS … common angle

 PQR SPR  (AA Test)

ii. Find the lengths of QR and PS.

Since ∆PQR SPR … from (i)

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models

iii.

Selina Solutions Icse Class 10 Mathematics Chapter - Similarity With Applications To Maps And Models