class 10 maths Triangles Hots Questions

Q 11 :

Two triangles are similar if their corresponding sides are __________.

equal
proportional
unequal
no relation

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(2)

Two triangles are similar if their corresponding sides are in proportion

Q 12 :

In Fig, in trapezium $A B C D i f A B ∥ C D,$ then the value of x is

29/8
8/29
20
1/20

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(3)

Since ABCD is a trapezium with AB ? CD and diagonals intersect at O

AO/OC = OB/OD ⇒ 2/5 = (x - 2)/(2x + 5)

= 2(2x + 5) = 5(x - 2)

= 4x + 10 = 5x - 10 $\Rightarrow$ x = 20

Option (c) is correct.

Q 13 :

In the given triangle (Right angled at C), DE || BC, AB = 5 cm, AD = 2 cm and BC = 3 cm. Using BPT/Thales theorem:

(i) AE = 8/5 cm

(ii) AE = 1 cm

(iii) AC = 4 cm

(iv) AC = 3 cm

Choose the correct option from the following:

(i) and (iii) are correct.
(i) and (iv) are correct.
(ii) and (iii) are correct.
(ii) and (iv) are correct

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(1)

Given right angled triangle ABC, using Pythagoras theorem we have

$A B ² = A C ² + B C ² \Rightarrow 5 ² = A C ² + 9 \Rightarrow A C ² = 25 - 9 = 16 \Rightarrow A C = 4 c m, H e n c e (i i i) i s c o r r e c t . G i v e n D E ∥ B C, u \sin g B P T t h e o r e m, A D / A B = A E / A C \Rightarrow 2 / 5 = A E / 4 \Rightarrow A E = 8 / 5 c m, H e n c e (i) i s c o r r e c t .$

Q 14 :

In the given triangle, we have AB = 12 cm, AC = 13 cm, AE = 13/4 cm, BC = 5 cm and AD = 3 cm. Then:

(i) ABC is a right-angled triangle

(ii) DE is parallel to BC

(iii) ABC is not a right-angled triangle

(iv) DE is perpendicular to AB

Choose the correct option from the following..

(i) and (iii) are correct.
(i) and (iv) are correct.
(i), (ii) and (iv) are correct.
Only (i) is correct.

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(4)

Checking for right-angled triangle:

$G i v e n, A B = 12 c m, B C = 5 c m a n d A C = 13 c m (A B) ² + (B C) ² = (12) ² + (5) ² = 144 + 25 = 169 = (13) ² S o, (A B) ² + (B C) ² = (A C) ², b y P y t h a g o r a s T h e o r e m, Δ A B C i s r i g h t a n g l e d a t B . H e n c e, (i) i s c o r r e c t, a n d (i i i) i s i n c o r r e c t N o w, A D / A B = 3 / 12 = 1 / 4 a n d A E / A C = (13 / 4 \times 1 / 13) = 1 / 4 B y B P T i f A D / A B = A E / A C, t h e n D E ∥ B C, H e n c e (i i) i s c o r r e c t . A s B i s a r i g h t a n g l e a n d D E ∥ B C, a n g l e D w i l l a l s o b e 90 °, s o D E ⟂ A B, H e n c e (i v) i s c o r r e c t .$

Q 15 :

In the given triangle ABC, PQ || BC. If PB = 6 cm, AP = 4 cm and AQ = 8 cm. Then

(i) AC = 20 cm

(ii) AC = 12 cm

(iii) AP, AQ and QC are in arithmetic progression

(iv) $A P / A B = A Q / A C$

Choose the correct option from the following

(i) and (iii) are correct.
(ii), (iii) and (iv) are correct
(i), (ii) and (iv) are correct.
(i), (iii) and (iv) are correct.

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(4)

$G i v e n P Q ∥ B C \Rightarrow A P / A B = A Q / A C (B y B P T) ∴ (i v) i s c o r r e c t . 4 / (4 + 6) = 8 / A C \Rightarrow A C = 20 c m, ∴ (i) i s c o r r e c t .$

QC = 20 − 8 = 12 cm

4, 8, 12 are in Arithmetic Progression.

Hence, (iii) is correct.

Q 16 :

$Δ A B C$ is right-angled at A and DEFG is a square as shown in the figure. Then:

$(i) Δ C B G ~ Δ E F C (i i) B D \times E C = D E ² (i i i) Δ A G F ~ Δ D B G (i v) Δ A G F ~ Δ E F C$

Choose the correct option from the following:

(iii) and (iv) are correct.
(i), (ii) and (iii) are correct.
(ii), (iii) and (iv) are correct.
(i) and (ii) are correct.

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(3)

$I n Δ A G F a n d Δ D B G : ∠ G A F = ∠ B D G = 90 ° a n d ∠ A G F = ∠ D B G (C o r r e s p o n d i n g a n g l e s) ∴ Δ A G F ~ Δ D B G (B y A A s i m i l a r i t y) \dots (A) ∴ (i i i) i s c o r r e c t . S i m i l a r l y, i n Δ A G F a n d Δ E F C : ∠ G A F = ∠ C E F = 90 ° a n d ∠ A F G = ∠ F C E (C o r r e s p o n d i n g a n g l e s) ∴ Δ A G F ~ Δ E F C (B y A A s i m i l a r i t y) \dots (B) ∴ (i v) i s c o r r e c t . F r o m (A) a n d (B) : Δ D B G ~ Δ E F C \frac{D B}{E F} = \frac{D G}{E C} B u t E F = D G = D E (s i d e o f s q u a r e) \frac{D B}{D E} = \frac{D E}{E C} \Rightarrow D E^{2} = D B \times E C ∴ (i i) i s c o r r e c t .$

Q 17 :

In the given figure, ABCD and AEFG are squares. Then

$(i) \frac{A F}{A G} = \frac{A C}{A D} (i i) \frac{A F}{A G} = \frac{A D}{A C} (i i i) Δ A C F ~ Δ A D G (i v) Δ A C F ~ Δ A G D$

Choose the correct option from the following:

(i) and (iii) are correct
(ii) and (iv) are correct
(ii) and (iii) are correct
(i) and (iv) are correct

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(1)

$I n Δ A G F a n d Δ A D C ∠ A G F = ∠ A D C (E a c h 90 °) ∠ G A F = ∠ D A C (E a c h 45 °) ∴ Δ A G F ~ Δ A D C (B y A A s i m i l a r i t y c r i t e r i o n) \Rightarrow \frac{A G}{A D} = \frac{A F}{A C} \Rightarrow \frac{A C}{A D} = \frac{A F}{A G} \dots (i) a l s o, ∠ D A C = ∠ G A F (E a c h 45 °) \Rightarrow ∠ D A C - ∠ G A C = ∠ G A F - ∠ G A C \Rightarrow ∠ D A G = ∠ C A F a n d \frac{A C}{A D} = \frac{A F}{A G} (F r o m (i)) ∴ Δ A C F ~ Δ A D G (B y S A S s i m i l a r i t y c r i t e r i o n)$

Topic Question Set

Q 11 :

Two triangles are similar if their corresponding sides are __________.

Q 12 :

In Fig, in trapezium $A B C D i f A B ∥ C D,$ then the value of x is

Q 13 :

In the given triangle (Right angled at C), DE || BC, AB = 5 cm, AD = 2 cm and BC = 3 cm. Using BPT/Thales theorem:

(i) AE = 8/5 cm

(ii) AE = 1 cm

(iii) AC = 4 cm

(iv) AC = 3 cm

Choose the correct option from the following:

Q 14 :

In the given triangle, we have AB = 12 cm, AC = 13 cm, AE = 13/4 cm, BC = 5 cm and AD = 3 cm. Then:

(i) ABC is a right-angled triangle

(ii) DE is parallel to BC

(iii) ABC is not a right-angled triangle

(iv) DE is perpendicular to AB

Choose the correct option from the following..

Q 15 :

In the given triangle ABC, PQ || BC. If PB = 6 cm, AP = 4 cm and AQ = 8 cm. Then

(i) AC = 20 cm

(ii) AC = 12 cm

(iii) AP, AQ and QC are in arithmetic progression

(iv) $A P / A B = A Q / A C$

Choose the correct option from the following

Q 16 :

$Δ A B C$ is right-angled at A and DEFG is a square as shown in the figure. Then:

$(i) Δ C B G ~ Δ E F C (i i) B D \times E C = D E ² (i i i) Δ A G F ~ Δ D B G (i v) Δ A G F ~ Δ E F C$

Choose the correct option from the following:

Q 17 :

In the given figure, ABCD and AEFG are squares. Then

$(i) \frac{A F}{A G} = \frac{A C}{A D} (i i) \frac{A F}{A G} = \frac{A D}{A C} (i i i) Δ A C F ~ Δ A D G (i v) Δ A C F ~ Δ A G D$

Choose the correct option from the following:

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Topic Question Set

Q 11 : Two triangles are similar if their corresponding sides are __________. .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

Q 12 : In Fig, in trapezium ABCD if AB ∥ CD, then the value of x is .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

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Q 11 :

Two triangles are similar if their corresponding sides are __________.

Q 12 :

In Fig, in trapezium $A B C D i f A B ∥ C D,$ then the value of x is