# RD SHARMA Solutions for Class 10 Maths Chapter 7 - Triangles

Page / Exercise

## Chapter 7 - Triangles Exercise Ex. 7.1

Question 1

Fill in the blanks using correct word given in the brackets:-
(i) All circles are __________. (congruent, similar)
(ii) All squares are __________. (similar, congruent)
(iii) All __________ triangles are similar. (isosceles, equilateral)

(iv) Two triangles are similar, if their corresponding angles are __________. (proportional, equal)

(v) Two triangles are similar, if their corresponding sides are __________. (proportional, equal)
(vi) Two polygons of the same number of sides are similar, if (a) their corresponding angles are __________ and (b) their corresponding sides are __________. (equal, proportional)

Solution 1

(i) All circles are similar.
(ii) All squares are similar.
(iii) All equilateral triangles are similar.

(iv) Two triangles are similar, if their corresponding angles are equal.

(v) Two triangles are similar, if their corresponding sides are proportional.
(vi) Two polygons of the same number of sides are similar, if (a) their corresponding angles are equal and (b) their corresponding sides are proportional.

Question 2

Write the truth value (T/F) of each of the following statements:

(i) Any two similar figures are congruent.

(ii) Any two congruent figures are similar.

(iii) Two polygons are similar, if their corresponding sides are proportional.

(iv) Two polygons are similar, if their corresponding angles are proportional.

(v) Two triangles are similar if their corresponding sides are proportional.

(vi) Two triangles are similar if their corresponding angles are proportional

Solution 2

(i) False

(ii) True

(iii) False

(iv) False

(v) True

(vi) True

## Chapter 7 - Triangles Exercise Ex. 7.2

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 1(ix)

Solution 1(ix)

Question 1(x)

Solution 1(x)

Question 1(xi)

Solution 1(xi)

Question 1(xii)

Solution 1(xii)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 3

Solution 3

Question 4

Solution 4

Question 5

In Fig 7.35, state if PQ || EF.

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

## Chapter 7 - Triangles Exercise Ex. 7.3

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 2

in Fig. 7.57, AE is the AE is the bisector of the exterior CAD Meeting BC produced in E. If AB = 10 cm, AC = 6 cm and BC = 12 cm, find CE.

Solution 2

Question 3

Solution 3

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 4(iii)

Solution 4(iii)

Question 4(iv)

Solution 4(iv)

Question 4(v)

Solution 4(v)

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

## Chapter 7 - Triangles Exercise Ex. 7.4

Question 1(i)

In fig., if AB||CD, find the value of x.

Solution 1(i)

Question 1(ii)

In fig., if AB || CD, find the value of x.

Solution 1(ii)

Question 1(iii)

In fig., AB||CD. If OA = 3x - 19, OB = x - 4, OC = x - 3 and OD = 4, find x.

Solution 1(iii)

## Chapter 7 - Triangles Exercise Ex. 7.5

Question 1

Solution 1

Question 2

In Fig. 7.137, AB || QR. Find the length of PB.

Solution 2

Question 3

In Fig. 7.138, XY || BC. Find the length of XY.

Solution 3

Question 4

In a right angled triangle with sides a and b and hypotenuse c, the altitude drawn on the hypotenuse is x. Prove that ab = cx.

Solution 4

We have:

Question 5

In Fig. 7.140, ABC = 90and BD AC. If BD = 8 cm and AD = 4 cm, find CD.

Solution 5

Question 6

In Fig. 7.140, ABC = 90o and BD AC> If AB = 5.7 cm , BD = 3.8 cm and CD = 5.4 cm, find BC.

Solution 6

Question 7

In fig. 7.141, DE || BC such that AE = (1/4) AC. If AB = 6 cm, find AD

Solution 7

Question 8

Solution 8

Question 9

Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using similarity criterion for two triangles, show that

Solution 9

Question 10

If ABC and AMP are two right triangles, right angled at B and M respectively such that MAP = BAC. Prove that

Solution 10

Question 11

A vertical stick 10 cm long casts a shadow 8 cm long. At the same time a tower casts a shadow 30 m long. Determine the height of the tower.

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

ABCD is a parallelogram and APQ is a straight line meeting BC at P and DC produced at Q. Prove that the rectangle obtained by BP and DQ is equal to the rectangle contained by AB and BC.

Solution 17

Question 18

In ABC, AL and CM are the perpendiculars from the vertices A anf C to BC and AB respectively. If AL and CM intersect at O, prove that:

(i)

(ii)

Solution 18

Question 19

ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the mid-points of AB, AC, CD and BD respectively, show that PQRS is a rhombus.

Solution 19

Question 20

In an isosceles ABC, the base AB is produced both the ways to P and Q such that AP BQ = AC2. Prove that .

Solution 20

Question 21

A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2m/sec. If the lamp is 3.6m above the ground, find the length of her shadow after 4 seconds.

Solution 21

Question 22

A vertical stick of a length 6 m casts a shadow 4m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.

Solution 22

Question 23

In fig. 7.144, ΔABC is right angled at C and DE AB. prove that ΔABC  ΔADE and hence find the lengths of AE and DE.

Solution 23

Question 24

Solution 24

Question 25

In Fig. 7.144, We have AB||CD||EF, if AB = 6 cm, CD = x cm, EF = 10 cm, BD = 4 cm and DE = y cm, calculate the values of x and y.

Solution 25

## Chapter 7 - Triangles Exercise Ex. 7.6

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 2

Solution 2

Question 3

The areas of two similar traingles are 81 cm2 and 49 cm2 respectively. Find the ratio of their corresponding heights. What is the ratio of their corresponding medians?

Solution 3

Question 4

The areas of two similar triangles are 169 cm2 and 121 cm2 respectively. If the longest side of the larger triangle is 26 cm, find the longest side of the smaller triangle.

?
Solution 4

Question 5

The areas of two similar triangles are 25 cm2 and 36 cm2 respectively. If the altitude of the first triangle is 2.4 cm, find the corresponding altitude of the other.

Solution 5

Question 6

The corresponding altitudes of two similar triangles are 6 cm and 9 cm respectively. Find the ratio of their areas.

Solution 6

Question 7

Solution 7

Question 8(i)

Solution 8(i)

Question 8(ii)

Solution 8(ii)

Question 8(iii)

Solution 8(iii)

Question 9

Solution 9

Question 10

Solution 10

Question 11

The areas of two similar triangles are 121 cm2 and 64 cm2 respectively. If the median of the first triangle is 12.1 cm, find the corresponding median of the other.

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 19

In Fig.7.180, ΔABC and ΔDBC are two triangles on the same base BC. If AD intersects BC at O,

show that

Solution 19

Since ABC and DBC are one same base,
Therefore ratio between their areas will be as ratio of their heights.
Let us draw two perpendiculars AP and DM on line BC.

In APO and DMO,
APO = DMO    (Each is90o)
AOP = DOM          (vertically opposite angles)
OAP = ODM         (remaining angle)
Therefore APO ~  DMO    (By AAA rule)

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 18

Diagonals of a trapezium PQRS intersect each other at the point O, PQ || RS and PQ = 3RS. Find the ratio of the areas of triangles POQ and ROS.

Solution 18

In trapezium PQRS, PQ || RS and PQ = 3RS.

… (i)

In POQ and ROS,

SOR = QOP … [Vertically opposite angles]

SRP = RPQ … [Alternate angles]

∴ ∆POQ ROS … [By AA similarity criteria]

Using the property of area of areas of similar triangles, we have

Hence, the ratio of the areas of triangles POQ and ROS is 9:1.

## Chapter 7 - Triangles Exercise Ex. 7.7

Question 1

Solution 1

Question 2(i)

The sides of a triangle are a = 7 cm, b = 24 cm and c = 25 cm. Determine whether it is a right triangle.

Solution 2(i)

Question 2(ii)

The sides of a triangle are a = 9 cm, b = 16 cm and c = 18 cm. Determine whether it is a right triangle.

Solution 2(ii)

Question 2(iii)

The sides of a triangle are a = 1.6 cm, b = 3.8 cm and c = 4 cm. Determine whether it is a right triangle.

Solution 2(iii)

Question 2(iv)

The sides of a triangle are a = 8 cm, b = 10 cm and c = 6 cm. Determine whether it is a right triangle.

Solution 2(iv)

Question 3

A man goes 15 metres due west and then 8 metres due north. How far is he from the starting point?

Solution 3

Question 4

A ladder 17 m long reaches a window of a building 15 m above the ground. Find the distance of the foot of the ladder from the building.

Solution 4

Question 5

Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distance between their tops.

Solution 5

Let CD and AB be the poles of height 11 and 6 m.
Therefore CP = 11 - 6 = 5 m
From the figure we may observe that AP = 12m
In triangle APC, by applying Pythagoras theorem

Therefore distance between their tops = 13 m.

Question 6

In an isosceles triangle ABC, AB = AC = 25 cm, BC = 14 cm. Calculate the altitude from A on BC.

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Using pythagoras theorem determine the length of AD terms of b and c shown in Fig. 7.221.

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

In Fig. 7.222, B<90o and segment AD BC, show that

Solution 17

(i)

Question 18

Solution 18

Question 19

ABD is a right triangle right angled at A and AC  BD. Show that
(i)    AB2 = BC . BD
(ii)    AC2 = BC . DC
(iii)    AD2 = BD . CD

(iv) AB2/ AC2 = BD/ DC

Solution 19

Question 20

A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?

Solution 20

Question 21

Determine whether the triangle having sides (a - 1) cm,  cm and (a + 1) cm is a right angled triangle.

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

In Fig. 7.223, D is the mid-point of side BC and AE  BC. If

Solution 24

Question 25

Solution 25

(i)

Question 26

Solution 26

Question 27

Solution 27

Question 28

An aeroplane leaves an airport and flies due north at a speed of 1,000 km per hour. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1,200 km per hour. How far apart will be the two planes after

Solution 28

## Chapter 7 - Triangles Exercise Rev. 7

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 2

Solution 2

Question 3

Solution 3

Question 4

In Fig. 7.226, DE || CB. Determine AC and AE

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7(i)

Solution 7(i)

Question 7(ii)

Solution 7(ii)

Question 7(iii)

Solution 7(iii)

Question 7(iv)

In the following figure, you find two triangles. Indicate whether the traingles are similar. Give reasons in support of your answer.

Solution 7(iv)

Incomplete question (two triangles are not given in the figure).

Question 7(v)

In the following figure, you find two triangles. Indicate whether the traingle are similar. Give reasons in support of your answer.

Solution 7(v)

Incomplete question (two triangles are not given in the figure).

Question 7(vi)

Solution 7(vi)

Question 8

Solution 8

Question 9

Solution 9

Question 10

In Fig. 7.229, Δ  CMD; determine MD in terms of x, y and z.

Solution 10

Question 11

Solution 11

Question 12

In Fig. 7.230, l || m

Solution 12

Question 13

In Fig. 7.231, AB || DC Prove that

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

In Fig. 7.232, each of PA , QB, RC and SD is perpendicular to l. if AB = 6 cm, BC = 9 cm, CD = 12 cm and PS = 36 cm, then determine PQ, QR and RS.

Solution 20

Question 21

In each of the figure given below, an altitude is drawn to the hypotenuse by a right-angled triangle. The length of different line-segments are marked in each figure. Determine x,y,z in each case.

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

There is a staircase as shown in Fig. 7.234, connecting points A and B. Measurements of steps are marketed in the figure. Find the straight line distance between A and B.

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Nazima is fly fishing in a stream. The tip of her fishing rod is 1.8 m above the surface of the water and the fly at the end of the string rests on the water 3.6 m away and 2.4 m from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taut, how much string does she have out (See fig.)? If she pulls the string at the rate of 5 cm per second, what will be the horizontal distance of the fly from her after 12 seconds.

Solution 38

The given information can be represented by the figure given below.

## Chapter 7 - Triangles Exercise 7.131

Question 1

Sides of two similar triangles are in the ratio 4:9.

Areas of these triangles are in the ratio

(a) 2:3        (b) 4:9        (c) 81:16         (d)16:81

Solution 1

We know if sides of two similar triangles are in ratio a:b then area of these triangles are in ratio a2b2

According to question, ratio of sides= 4:9

Hence ratio of areas = 42:92

= 16:81

So, the correct option is (d).

Question 2

The areas of two similar triangles are in respectively 9 cm2 and 16 cm2. The ratio of this corresponding sides is

(a) 3:4        (b) 4:3          (c)2:3    (d) 4:5

Solution 2

So, the correct option is (a).

Question 3

The areas of two similar triangles ∆ ABC and ∆ DEF are 144 cm2 and 81 cm2 respectively. If the longest side of larger ∆ABC be 36cm, then longest side of smaller triangle ∆DEF is

(a) 20 cm       (b) 26 cm      (c)27 cm       (d) 30 cm

Solution 3

## Chapter 7 - Triangles Exercise 7.132

Question 4

∆ABC and ∆BDE are two equilateral triangles such that D is the mid-point of BC. The ratio of areas of triangles ABC and BDE is

(a) 2:1        (b) 1:2         (c) 4:1         (d)1:4

Solution 4

Question 5

If ∆ABC and ∆DEF are similar such that 2AB+DE and BC=8 cm, then EF=

(a) 16 cm     (b)12 cm    (c) 8 cm     (d) 4 cm

Solution 5

Question 6

Solution 6

All these pairs of corresponding sides are in the same proportion so by SSS similarity criteria triangle ∆ABC are similar.

Given ratio of sides = 2.5

So, ratio of areas    = 22:52

= 4:25

So, the correct option is (b).

Question 7

XY is drawn parallel to the base BC of a ∆ABC cutting AB at x and X and AC at Y. If AB=4BX  and YC=2 cm

Then AY=

(a) 2 cm        (b) 4 cm        (c) 6 cm        (d) 8 cm

Solution 7

Question 8

Two poles of height 6 m and 11 m stand vertically upright on a plane ground. If the distance between their foot is 12 m, the distance between their tops is

(a) 12 m        (b)14 m        (c)13 m    (d)11 m

Solution 8

Question 9

In ∆ABC, D and E are points on side AB and AC respectively such that DE∥BC and AD:DB=3:1

If EA = 3.3 cm, then AC=

(a) 1.1 cm       (b) 4 cm          (c) 4.4 cm    (d) 5. 5 cm

Solution 9

Question 10

In triangles ABC and DEF, ∠A =∠E=40°, AB:ED= AC : EF and ∠F = 65°, then ∠B=

(a) 35°       (b) 65°       (c) 75°      (d) 85°

Solution 10

Question 11

If ABC and DEF are similar triangles such that ∠A=47° and ∠E = 83°, then ∠C=

(a)50°            (b)60°            (c)70°        (d)80°

Solution 11

Question 12

If D, E, F are the mid-points of sides BC, CA and AB respectively of ∆ABC, then the ratio of areas of triangles DEF and ABC is

(a) 1:4          (b) 1:2          (c) 2:3        (d)4:5

Solution 12

Question 13

In an equilateral triangle ABC, if AD ⏊ BC, then

Solution 13

Question 14

Solution 14

Question 15

In a ∆ABC, AD is the bisector of ∠BAC. If AB = 6 cm,

AC=5 cm  and BD=3 cm, then DC=

(a) 11.3 cm    (b)2.5cm    (c) 3.5 cm   (d) None of these

Solution 15

Question 16

In a ∆ABC, AD is bisector of ∠BAC. If AB=8cm, BD=6cm,  and DC=3cm, Find AC

(a) 4cm     (b)6cm      (c)3cm       (d)8cm

Solution 16

Question 17

Solution 17

## Chapter 7 - Triangles Exercise 7.133

Question 18

If ABC is a right triangle, right angled at B and M, N are mid points of AB and BC respectively, then 4(AN2+CM2)=

(a) 4 AC2      (b) 5 AC2      (a) 4 AC2        (c) 4 AC2

Solution 18

Question 19

Solution 19

For triangles to be similar by SAS

∠B = ∠D

So, the correct option is (c).

Question 20

Solution 20

Question 21

Solution 21

So, the correct option is (a).

Question 22

Solution 22

Question 23

A man goes 24 m due west and then 7 m due north.

How far is he from the starting point?

(a)31m       (b)17m       (c)25m       (d)26m

Solution 23

Question 24

Solution 24

So, the correct option is (b).

Question 25

Solution 25

So, the correct option is (c).

Question 26

The areas of two similar triangles are 121cm2 and 64cm2 respectively. If the median of the first triangle is 12.1cm, then the corresponding median of the other triangle is

(a) 11cm    (b)8.8cm    (c)11.1cm   (d)8.1cm

Solution 26

Question 27

(a) CD2       (b) 2CD2       (c) 3CD2     (d) 4CD2

Solution 27

So, the correct option is (c).

Question 28

In an equilateral triangle ABC if AD⊥ BC, then

Solution 28

So, the correct option is (b).

Question 29

In an isosceles triangle ABC if AC=BC and AB2=2AC2 then ∠C

(a) 300      (b)450        (c)900           (d)600

Solution 29

Question 30

Solution 30

## Chapter 7 - Triangles Exercise 7.134

Question 31

Solution 31

So, the correct option is (b).

Question 32

In fig, measures of ∠D and ∠F are respectively

(a) 50°, 40°   (b) 20°, 30°    (c) 40°, 50°    (d) 30°, 30°

Solution 32

Question 33

In fig, the value of x for which DE||BC is

(a) 4     (b)1    (c)3    (d)2

Solution 33

Question 34

In fig, if ∠ADE = ∠ABC, then CE=

(a) 2      (b)5     (c)     (d)3

Solution 34

Question 35

Solution 35

## Chapter 7 - Triangles Exercise 7.135

Question 36

Solution 36

Question 37

Solution 37

Question 38

A vertical stick 20 m long cast a shadow 10 m long on the ground. At the same time, a tower casts a shadow 50 m long on the ground. The height of tower is

(a) 100 m   (b) 120 m   (c) 25 m   (d) 200 m

Solution 38

Question 39

Two isosceles triangles have equal angles and their areas are in ratio 16:25. The ratio of their corresponding heights is

(a) 4:5            (b) 5:4         (c) 3:2           (d) 5:7

Solution 39

So, the correct option is (a).

Question 40

∆ABC is such that AB=3 cm, BC=2 cm and CA=2.5 cm.

If ∆DEF ∿ ∆ABC and EF=4cm, then perimeter of ∆DEF is

(a) 7.5cm       (b) 15cm      (c) 22.5cm      (d) 30 cm

Solution 40

So, the correct option is (b).

## Chapter 7 - Triangles Exercise 7.136

Question 41

In ∆ABC, a line XY parallel to BC cuts AB at X and AC at Y.

If BY bisects ∠XYC, then

(a) BC=CY      (b) BC=BY     (c)BC≠CY   (d) BC≠BY

Solution 41

Question 42

Solution 42

Question 43

In a ∆ABC, perpendicular AD from A on BC meets BC at D. If BD=8 cm,  DC=2cm  and AD=4cm  then

(a) ∆ABC is isosceles      (b) ∆ABC is equilateral

(c) AC = 2AB                (d) ∆ABC is right angled at A

Solution 43

Question 44

Solution 44

Question 45

If ABC is an isosceles triangle and D is a point on BC such that AD⊥ BC, then

Solution 45

Question 46

Solution 46

Question 47

If E is a point on side CD of an equilateral triangle ABC such that BE ⊥CA, then AB2+BC2+CA2=

(a) 2BE2      (b) 3BE2      (c) 4BE2     (d) 2BE2

Solution 47

Question 48

Solution 48

Question 49

If ∆ABC ∿∆DEF  such that DE=3cm, EF=2cm, DF=2.5cm, BC=4cm then perimeter of ∆ABC is

(a)18cm    (b)20cm    (c)12cm   (d)15cm

Solution 49

Question 50

If ∆ABC ∿ ∆DEF such that AB=9.1 cm and DE=6.5. If the perimeter of ∆DEF is 25 cm, then the perimeter of ∆ABC is

(a) 36 cm   (b) 30 cm   (c) 34 cm   (d)35cm

Solution 50

Question 51

In an isosceles triangle ABC, if AB=AC=25 cm and BC=14 cm, then the measure of altitude from A on BC is

(a) 20cm  (b) 22cm     (c) 18 cm    (d)24 cm

Solution 51

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