Class 10 Maths Triangles questions with solutions

Q 1 :

The discus throw is an event in which an athlete aims to throw a discus as far as possible. The athlete spins counterclockwise one and a half times within a circle before releasing the discus. Upon release, the discus travels along a tangent to the circular path of the spin.

In the given figure, AB is one such tangent to a circle of radius 75 cm. Point O is centre of the circle and $∠ A B O = 30 ° .$ PQ is parallel to OA

Based on the above information answer the following questions:

(i) Find the length of AB.

$75 \sqrt{3} cm$
$75 \sqrt{2} cm$
$65 \sqrt{3} cm$
$- 75 \sqrt{3} cm$

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(1)

Since OA is radius and BA is tangent to the circle at point A

$∴ ∠ O A B = 90 ° I n Δ O A B, \tan 30 ° = \frac{O A}{A B} \Rightarrow \frac{1}{\sqrt{3}} = \frac{75}{A B} \Rightarrow A B = 75 \sqrt{3} c m$

Q 2 :

The discus throw is an event in which an athlete aims to throw a discus as far as possible. The athlete spins counterclockwise one and a half times within a circle before releasing the discus. Upon release, the discus travels along a tangent to the circular path of the spin.

In the given figure, AB is one such tangent to a circle of radius 75 cm. Point O is centre of the circle and $angle A B O space equals space 30 degree. space$ PQ is parallel to OA

Based on the above information answer the following questions:

(ii) Find the length of OB.

150 cm
140
130
120

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(1)

$I n Δ O A B, w e h a v e \sin 30 ° = \frac{O A}{O B} \Rightarrow \frac{1}{2} = \frac{75}{O B} \Rightarrow O B = 150 cm$

Q 3 :

The discus throw is an event in which an athlete aims to throw a discus as far as possible. The athlete spins counterclockwise one and a half times within a circle before releasing the discus. Upon release, the discus travels along a tangent to the circular path of the spin.

In the given figure, AB is one such tangent to a circle of radius 75 cm. Point O is centre of the circle and $angle A B O space equals space 30 degree. space$ PQ is parallel to OA

Based on the above information answer the following questions:

(iii) (a) Find the length of AP

$\frac{75 \sqrt{3}}{2} cm$
$\frac{75 \sqrt{4}}{2} cm$
$- \frac{75 \sqrt{3}}{4} cm$
$\frac{76 \sqrt{3}}{2} cm$

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(1)

Since the radius of circle is perpendicular to the tangent at the point of contact $∴ ∠ O A P = 90 °$

$S i n c e P Q ∥ O A \Rightarrow ∠ O A P = ∠ Q P B = 90 ° (C o r r e s p o n d i n g a n g l e s) N o w, O B = 150 c m O Q + Q B = 150 \Rightarrow 75 + Q B = 150 \Rightarrow Q B = 150 - 75 = 75 cm I n r i g h t Δ P B Q, \cos 30 ° = \frac{P B}{Q B} \Rightarrow \frac{\sqrt{3}}{2} = \frac{P B}{75} \Rightarrow P B = \frac{75 \sqrt{3}}{2} cm A P = A B - P B = 75 \sqrt{3} - \frac{75 \sqrt{3}}{2} \Rightarrow A P = \frac{75 \sqrt{3}}{2} cm$

Q 4 :

The discus throw is an event in which an athlete aims to throw a discus as far as possible. The athlete spins counterclockwise one and a half times within a circle before releasing the discus. Upon release, the discus travels along a tangent to the circular path of the spin.

In the given figure, AB is one such tangent to a circle of radius 75 cm. Point O is centre of the circle and $angle A B O space equals space 30 degree. space$ PQ is parallel to OA

Based on the above information answer the following questions:

(iii) (b) Find the length of PQ.

35.5
37.5
36.5
33.5

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(2)

$I n r i g h t Δ Q P B, \sin \ funcapply 30 ° = \frac{P Q}{Q B} \Rightarrow \frac{1}{2} = \frac{P Q}{75} \Rightarrow P Q = \frac{75}{2} = 37.5 cm$

Q 5 :

A farmer had a triangular piece of land. He put a fence, parallel to one of the sides of the field as shown in the figure.

Based on the above information answer the following questions:

(i) If the point D is 20 m away from point A, whereas AB and AC are 80 m and 100 m respectively, find the length of AE.

20 m
15 m
25 m
10 m

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(3)

$I n Δ A B C, D E ∥ B C \frac{A D}{A B} = \frac{A E}{A C} (By BPT) \frac{20}{80} = \frac{A E}{100} \Rightarrow A E = 25 m$

Q 6 :

A farmer had a triangular piece of land. He put a fence, parallel to one of the sides of the field as shown in the figure.

Based on the above information answer the following questions:

(ii) If AD = x + 1, DB = 3x – 1, AE = x + 3, EC = 3x + 4, then find the value of x.

7
5
4
3

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4

(1)

Given, AD = x + 1, DB = 3x – 1, AE = x + 3, EC = 3x + 4.

$S i n c e, i n Δ A B C, D E ∥ B C \frac{A D}{D B} = \frac{A E}{E C} (Using BPT) \frac{x + 1}{3 x - 1} = \frac{x + 3}{3 x + 4} C r o s s m u l t i p l y i n g : (x + 1) (3 x + 4) = (3 x - 1) (x + 3) 3 x^{2} + 7 x + 4 = 3 x^{2} + 8 x - 3 7 x + 4 = 8 x - 3 \Rightarrow x = 7$

Q 7 :

A farmer had a triangular piece of land. He put a fence, parallel to one of the sides of the field as shown in the figure.

Based on the above information answer the following questions:

(iii) (a) State whether $\frac{B D}{A D} = \frac{A E}{E C}$ or not. Give reason

$\frac{B D}{A D} \neq \frac{A E}{E C}$
$\frac{A D}{D B} \neq \frac{A E}{E C}$
$\frac{B D}{A E} \neq \frac{D E}{E C}$
$\frac{B D}{A D} = \frac{A E}{E C}$

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(1)

$N o, b e c a u s e w h e n i n Δ A B C, D E ∥ B C \frac{A D}{D B} = \frac{A E}{E C} (Using BPT) \Rightarrow \frac{1}{A D / D B} = \frac{1}{A E / E C} \Rightarrow \frac{B D}{A D} = \frac{E C}{A E} H e n c e, \frac{B D}{A D} \neq \frac{A E}{E C}$

Q 8 :

A farmer had a triangular piece of land. He put a fence, parallel to one of the sides of the field as shown in the figure.

Based on the above information answer the following questions:

(iii) (b) If P and Q are the midpoints of sides YZ and XZ respectively in $Δ Z Y X$ , then state whether $P Q ∥ Y X$ or not. Give reason.

$P Q ∥ Y X$
$P X ∥ Y Q$
$Q X ∥ Y P$
$- P X ∥ Y Q$

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(1)

$I n Δ Z Y X,$ P and Q are midpoints of side YZ and XZ respectively.

$\frac{Z P}{P Y} = 1 and \frac{Z Q}{Q X} = 1 \Rightarrow \frac{Z P}{P Y} = \frac{Z Q}{Q X} \Rightarrow P Q ∥ Y X (By converse of BPT) Y e s, P Q ∥ Y X .$

Q 9 :

Vijay is trying to estimate the height of a tower near his house using the properties of similar triangles. His house, which is 20 m tall, casts a 10 m long shadow on the ground.

At the same time, the tower casts a 50 m long shadow on the ground, and Ajay’s house casts a 20 m long shadow on the ground.

Based on the above information, answer the following questions:

(i) What is the height of the tower?

100 m
150 m
120 m
140 m

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(1)

The figure can be drawn as Fig. 6.35.

Let CD = h m be the height of the tower, BE = 20 m be the height of Vijay’s house, and GF be the height of Ajay’s house.

Since, $∆ A C D ~ ∆ A B E \frac{A C}{A B} = \frac{C D}{E B} \frac{50}{10} = \frac{h}{20} h = 100 m$

Q 10 :

Vijay is trying to estimate the height of a tower near his house using the properties of similar triangles. His house, which is 20 m tall, casts a 10 m long shadow on the ground.

At the same time, the tower casts a 50 m long shadow on the ground, and Ajay’s house casts a 20 m long shadow on the ground.

Based on the above information, answer the following questions:

(ii) What will be the length of the shadow of the tower when Vijay’s house casts a 12 m long shadow?

50 m
60 m
40 m
30 m

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(2)

Given AB = 12 m, let AC = x.

$I n s i m i l a r Δ A B E a n d Δ A C D : \frac{A B}{A C} = \frac{B E}{C D} \frac{12}{x} = \frac{20}{100} x = \frac{12 \times 100}{20} = 12 \times 5 = 60 m$

Q 11 :

Vijay is trying to estimate the height of a tower near his house using the properties of similar triangles. His house, which is 20 m tall, casts a 10 m long shadow on the ground.

At the same time, the tower casts a 50 m long shadow on the ground, and Ajay’s house casts a 20 m long shadow on the ground.

Based on the above information, answer the following questions:

(iii) (a) What is the height of Ajay’s house?

40 m
20 m
25 m
30 m

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(1)

Let height of Ajay’s house be $G F = h ₁ .$

Since:

$∆ H F G ~ ∆ H C D \frac{H F}{H C} = \frac{F G}{C D} \frac{20}{50} = \frac{h_{1}}{100} h_{1} = \frac{20 \times 100}{50} = 40 m$

Q 12 :

Vijay is trying to estimate the height of a tower near his house using the properties of similar triangles. His house, which is 20 m tall, casts a 10 m long shadow on the ground.

At the same time, the tower casts a 50 m long shadow on the ground, and Ajay’s house casts a 20 m long shadow on the ground.

Based on the above information, answer the following questions:

(iii) (b) When the tower casts a 40 m long shadow, what will be the length of the shadow cast by Ajay’s house at the same time?

13 m
14 m
11 m
16 m

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(4)

Given, HC = 40 m.
Let length of the shadow of Ajay’s house be HF = l m.

Since:

$∆ H F G ~ ∆ H C D \frac{H F}{H C} = \frac{F G}{C D} \frac{l}{40} = \frac{40}{100} l = \frac{40 \times 40}{100} = 16 m$

Q 13 :

The Burj Khalifa, located in Dubai, United Arab Emirates, is the tallest tower in the world, standing at approximately 828 m. It features the highest observation deck open to the public. While walking on the deck, a person observed the shadows of the Burj Khalifa and nearby buildings. At one point, he noted that the shadow of the Burj Khalifa measured 207 m, while the shadow of a building named ‘P’ was 46 m long.

Based on the above information answer the following questions:

(i) Name the property which can be used to find out the length of the building ‘P’.

CD
DM
AB
BD

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(1)

$(i) Δ A B M ~ Δ C D M (B y A A s i m i l a r i t y) H e n c e, \frac{D M}{B M} = \frac{C D}{A B} (by CPCT)$

Using this we can find the height of the building P, i.e. CD

Q 14 :

The Burj Khalifa, located in Dubai, United Arab Emirates, is the tallest tower in the world, standing at approximately 828 m. It features the highest observation deck open to the public. While walking on the deck, a person observed the shadows of the Burj Khalifa and nearby buildings. At one point, he noted that the shadow of the Burj Khalifa measured 207 m, while the shadow of a building named ‘P’ was 46 m long.

Based on the above information answer the following questions:

(ii) Calculate the height of building ‘P’.

180
184
175
189

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4

(2)

$∵ Δ A B M ~ Δ C D M ∴ \frac{A B}{C D} = \frac{B M}{D M} \Rightarrow \frac{828}{P} = \frac{207}{46} \Rightarrow P = \frac{828 \times 46}{207} = 4 \times 46 = 184 m$

Q 15 :

The Burj Khalifa, located in Dubai, United Arab Emirates, is the tallest tower in the world, standing at approximately 828 m. It features the highest observation deck open to the public. While walking on the deck, a person observed the shadows of the Burj Khalifa and nearby buildings. At one point, he noted that the shadow of the Burj Khalifa measured 207 m, while the shadow of a building named ‘P’ was 46 m long.

Based on the above information answer the following questions:

(iii) (a) What is the length of shadow of Burj Khalifa when the length of shadow of building ‘Q’ is 81 metres?

355.5
345.6
364.5
360.2

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(3)

When length of shadow of building Q = 81 m = l, let length of the shadow of Burj Khalifa at the instant = x m.

$\frac{828}{P} = \frac{x}{l} \Rightarrow \frac{828}{184} = \frac{x}{81} \Rightarrow x = \frac{81 \times 828}{184} = 364.5 m$

Q 16 :

The Burj Khalifa, located in Dubai, United Arab Emirates, is the tallest tower in the world, standing at approximately 828 m. It features the highest observation deck open to the public. While walking on the deck, a person observed the shadows of the Burj Khalifa and nearby buildings. At one point, he noted that the shadow of the Burj Khalifa measured 207 m, while the shadow of a building named ‘P’ was 46 m long.

Based on the above information answer the following questions:

(iii) (b) At the same instance when the length of the shadow of Burj Khalifa was 207 m, what will be the length of the shadow of building ‘Q’ of height 108 m?

27
30
25
32

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(1)

We have,

$\frac{828}{108} = \frac{207}{Length of shadow of building Q} \Rightarrow Length of shadow of building Q = \frac{207 \times 108}{828} = 27 m$

Q 17 :

The law of reflection states that when a ray of light reflects off a surface, the angle of incidence is equal to the angle of reflection.

Ramesh places a mirror on level ground to determine the height of a pole. He stands at a certain distance so that he can see the top of the pole reflected in the mirror. Ramesh’s eye level is 1.5 m above the ground. The distances from the mirror to Ramesh and from the mirror to the pole are 1.8 m and 6 m, respectively.

Based on the above information, answer the following questions:

(i) Which criterion of similarity is applicable to the formed triangles?

AA
AB
BC
BD

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(1)

From the Fig. 6.109,

$∠ A B M = ∠ C D M and ∠ A M B = ∠ C M D ∴ Δ A B M ~ Δ C D M (By AA similarity)$

Thus, AA similarity criterion is applicable to the triangles

Q 18 :

The law of reflection states that when a ray of light reflects off a surface, the angle of incidence is equal to the angle of reflection.

Ramesh places a mirror on level ground to determine the height of a pole. He stands at a certain distance so that he can see the top of the pole reflected in the mirror. Ramesh’s eye level is 1.5 m above the ground. The distances from the mirror to Ramesh and from the mirror to the pole are 1.8 m and 6 m, respectively.

Based on the above information, answer the following questions:

(ii) What is the height of the pole?

5 m
4m
3m
2m

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(1)

We have,

$\frac{h}{1.5} = \frac{6}{1.8} (Since Δ A B M ~ Δ C D M) \Rightarrow \frac{h}{1.5} = \frac{6}{1.8} = \frac{1}{0.3} = \frac{1}{0.3} \Rightarrow \frac{h}{1.5} = \frac{1}{0.3} \Rightarrow \frac{h}{15} = \frac{6}{18} \Rightarrow \frac{h}{15} = \frac{1}{3} \Rightarrow h = \frac{15}{3} = 5 m So, the height of the pole is 5 m .$

Q 19 :

The law of reflection states that when a ray of light reflects off a surface, the angle of incidence is equal to the angle of reflection.

Ramesh places a mirror on level ground to determine the height of a pole. He stands at a certain distance so that he can see the top of the pole reflected in the mirror. Ramesh’s eye level is 1.5 m above the ground. The distances from the mirror to Ramesh and from the mirror to the pole are 1.8 m and 6 m, respectively.

Based on the above information, answer the following questions:

(iii) (a) What is the distance between mirror and pole?

6 m
5 m
4 m
3 m

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4

(1)

Distance between mirror and pole is 6 m.

Q 20 :

The law of reflection states that when a ray of light reflects off a surface, the angle of incidence is equal to the angle of reflection.

Ramesh places a mirror on level ground to determine the height of a pole. He stands at a certain distance so that he can see the top of the pole reflected in the mirror. Ramesh’s eye level is 1.5 m above the ground. The distances from the mirror to Ramesh and from the mirror to the pole are 1.8 m and 6 m, respectively.

Based on the above information, answer the following questions:

(iii) (b) Now, Ramesh moves back so that the distance between him and the pole is 13 m. He places the mirror between himself and the pole to see the reflection of the pole's top correctly. What is the distance between Ramesh and the mirror?

3m
4m
5m
2m

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(1)

Let MD = x m

Then $B M = (13 - x) m$

$N o w, \sin c e Δ A B M ~ Δ C D M, \frac{A B}{C D} = \frac{B M}{D M} \Rightarrow \frac{5}{1.5} = \frac{13 - x}{x} \frac{5}{1.5} = \frac{13 - x}{x} \Rightarrow \frac{50}{15} = \frac{13}{x} - 1 \Rightarrow \frac{13}{x} = 1 + \frac{10}{3} = \frac{13}{3} \Rightarrow \frac{13}{x} = \frac{13}{3} \Rightarrow x = 3 m$

Thus, distance between mirror and Ramesh is 3 m.

Topic Question Set

Q 6 :

A farmer had a triangular piece of land. He put a fence, parallel to one of the sides of the field as shown in the figure.

Based on the above information answer the following questions:

(ii) If AD = x + 1, DB = 3x – 1, AE = x + 3, EC = 3x + 4, then find the value of x.

Q 7 :

A farmer had a triangular piece of land. He put a fence, parallel to one of the sides of the field as shown in the figure.

Based on the above information answer the following questions:

(iii) (a) State whether $\frac{B D}{A D} = \frac{A E}{E C}$ or not. Give reason

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Topic Question Set

Q 6 : A farmer had a triangular piece of land. He put a fence, parallel to one of the sides of the field as shown in the figure. Based on the above information answer the following questions: (ii) If AD = x + 1, DB = 3x – 1, AE = x + 3, EC = 3x + 4, then find the value of x.

Q 7 : A farmer had a triangular piece of land. He put a fence, parallel to one of the sides of the field as shown in the figure. Based on the above information answer the following questions: (iii) (a) State whether BDAD=AEEC or not. Give reason

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Q 6 :

A farmer had a triangular piece of land. He put a fence, parallel to one of the sides of the field as shown in the figure.

Based on the above information answer the following questions:

(ii) If AD = x + 1, DB = 3x – 1, AE = x + 3, EC = 3x + 4, then find the value of x.

Q 7 :

A farmer had a triangular piece of land. He put a fence, parallel to one of the sides of the field as shown in the figure.

Based on the above information answer the following questions:

(iii) (a) State whether $\frac{B D}{A D} = \frac{A E}{E C}$ or not. Give reason