RD SHARMA SOLUTION CHAPTER- 4 TRIANGLES| CLASS 10TH MATHEMATICS-EDUGROWN

Chapter 4 – Triangles Exercise Ex. 4.1

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 4 – Triangles Exercise Ex. 4.2

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 4 – Triangles Exercise Ex. 4.3

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 4 – Triangles Exercise Ex. 4.4

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 4 – Triangles Exercise Ex. 4.5

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 = 90^o and BD AC. If BD = 8 cm and AD = 4 cm, find CD.

Solution 5

Question 6

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.

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 4 – Triangles Exercise Ex. 4.6

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 is90^o)
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 4 – Triangles Exercise Ex. 4.7

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<90^o 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/ DCSolution 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