RD SHARMA Solutions for Class 10 Maths Chapter 7 - Triangles
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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 = 90o and 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 18
Two isosceles triangles have equal vertical angles and their areas area in the ratio 36:25. Find the ratio of their corresponding heights.
Solution 18
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
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
(a) 2AB2=3AD2 (b) 4 AB2 =3AD2
(c) 3AB2=4AD2 (d) 3AB2 = 2AD2
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
In an equilateral triangle ABC if AD⊥ BC, then AD2
(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
(a) 5AB2= 4AD2 (b) 3AB2= 4AD2
(c) 4AB2= 3AD2 (d) 2AB2= 3AD2
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
(a)AB2-AD2=BD.DC (b) AB2-AD2=BD2-DC2
(c) AB2+AD2=BD.DC (d) AB2+AD2=BD2-DC2
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
Other Chapters for CBSE Class 10 Mathematics
Chapter 1- Real Numbers Chapter 2- Polynomials Chapter 3- Pairs of Linear Equations in Two Variables Chapter 4- Quadratic Equations Chapter 5- Arithmetic Progressions Chapter 6- Co-ordinate Geometry Chapter 8- Circles Chapter 9- Constructions Chapter 10- Trigonometric Ratios Chapter 11- Trigonometric Identities Chapter 12- Heights and Distances Chapter 13- Areas Related to Circles Chapter 14- Surface Areas and Volumes Chapter 15- Statistics Chapter 16- ProbabilityRD SHARMA Solutions for CBSE Class 10 Subjects
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