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

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## Chapter 7 - Triangles Exercise 7.131

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

We know if sides of two similar triangles are in ratio a:b then area of these triangles are in ratio a^{2}b^{2}

According to question, ratio of sides= 4:9

Hence ratio of areas = 4^{2}:9^{2}

= 16:81

So, the correct option is (d).

The areas of two similar triangles are in respectively 9 cm^{2} and 16 cm^{2}. The ratio of this corresponding sides is

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

So, the correct option is (a).

The areas of two similar triangles ∆ ABC and ∆ DEF are 144 cm^{2} and 81 cm^{2} 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

## Chapter 7 - Triangles Exercise 7.132

∆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

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

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 = 2^{2}:5^{2 }

= 4:25

So, the correct option is (b).

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

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

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

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°

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

(a)50° (b)60° (c)70°^{ } (d)80°

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

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

(a) 2AB^{2}=3AD^{2} (b) 4 AB^{2} =3AD^{2}

(c) 3AB^{2}=4AD^{2} (d) 3AB^{2} = 2AD^{2}

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

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

## Chapter 7 - Triangles Exercise 7.133

If ABC is a right triangle, right angled at B and M, N are mid points of AB and BC respectively, then 4(AN^{2}+CM^{2})=

(a) 4 AC^{2} (b) 5 AC^{2} (a) 4 AC^{2} (c) 4 AC^{2}

For triangles to be similar by SAS

∠B = ∠D

So, the correct option is (c).

So, the correct option is (a).

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

So, the correct option is (b).

So, the correct option is (c).

The areas of two similar triangles are 121cm^{2} and 64cm^{2} 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

In an equilateral triangle ABC if AD⊥ BC, then AD^{2}

(a) CD^{2} (b) 2CD^{2 }(c) 3CD^{2 }(d) 4CD^{2}

So, the correct option is (c).

In an equilateral triangle ABC if AD⊥ BC, then

(a) 5AB^{2}= 4AD^{2} (b) 3AB^{2}= 4AD^{2}

(c) 4AB^{2}= 3AD^{2} (d) 2AB^{2}= 3AD^{2}

So, the correct option is (b).

In an isosceles triangle ABC if AC=BC and AB^{2}=2AC^{2} then ∠C

(a) 30^{0} (b)45^{0} (c)90^{0} (d)60^{0}

## Chapter 7 - Triangles Exercise 7.134

So, the correct option is (b).

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

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

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

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

In fig, if ∠ADE = ∠ABC, then CE=

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

## Chapter 7 - Triangles Exercise 7.135

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

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

So, the correct option is (a).

∆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

So, the correct option is (b).

## Chapter 7 - Triangles Exercise 7.136

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

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

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

(a)AB^{2}-AD^{2}=BD.DC (b) AB^{2}-AD^{2}=BD^{2}-DC^{2}

(c) AB^{2}+AD^{2}=BD.DC (d) AB^{2}+AD^{2}=BD^{2}-DC^{2}

If E is a point on side CD of an equilateral triangle ABC such that BE ⊥CA, then AB^{2}+BC^{2}+CA^{2}=

(a) 2BE^{2} (b) 3BE^{2} (c) 4BE^{2 }(d) 2BE^{2}

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

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

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

## Chapter 7 - Triangles Exercise Ex. 7.1

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)

(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.

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

(i) False

(ii) True

(iii) False

(iv) False

(v) True

(vi) True

## Chapter 7 - Triangles Exercise Ex. 7.2

In Fig 7.35, state if PQ || EF.

## Chapter 7 - Triangles Exercise Ex. 7.3

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.

## Chapter 7 - Triangles Exercise Ex. 7.4

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

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

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

## Chapter 7 - Triangles Exercise Ex. 7.5

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

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

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.

**We have: **

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

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

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

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

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

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.

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.

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)** **

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.

In an isosceles ABC, the base AB is produced both the ways to P and Q such that AP BQ = AC^{2}. Prove that .

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.

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.

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.

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.

## Chapter 7 - Triangles Exercise Ex. 7.6

**The areas of two similar traingles are 81 cm ^{2} and 49 cm^{2} respectively. Find the ratio of their corresponding heights. What is the ratio of their corresponding medians?**

**The areas of two similar triangles are 169 cm ^{2} and 121 cm^{2} respectively. If the longest side of the larger triangle is 26 cm, find the longest side of the smaller triangle.**

**?**

The areas of two similar triangles are 25 cm^{2} and 36 cm^{2} respectively. If the altitude of the first triangle is 2.4 cm, find the corresponding altitude of the other.

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

The areas of two similar triangles are 121 cm^{2} and 64 cm^{2} respectively. If the median of the first triangle is 12.1 cm, find the corresponding median of the other.

Two isosceles triangles have equal vertical angles and their areas area in the ratio 36:25. Find the ratio of their corresponding heights.

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

show that** **

** **

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 is90^{o})

AOP = DOM (vertically opposite angles)

OAP = ODM (remaining angle)

Therefore APO ~ DMO (By AAA rule)

## Chapter 7 - Triangles Exercise Ex. 7.7

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

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

**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.**

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

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

**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. **

**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.**

** ****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.**

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

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

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

**(i)**

ABD is a right triangle right angled at A and AC BD. Show that

(i) AB^{2} = BC . BD

(ii) AC^{2} = BC . DC

(iii) AD^{2} = BD . CD

(iv) AB^{2}/ AC^{2} = BD/ DC

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?

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

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

**(i) **

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

## Chapter 7 - Triangles Exercise Rev. 7

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

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

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

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

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

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

In Fig. 7.230, l || m

In Fig. 7.231, AB || DC Prove that

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.

**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.**

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.

**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. **

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

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