In an equilateral triangle A B C , D is a point on side B C such that B D = 1 3 B C . Prove that 9 A D 2 = 7 A B 2 .

Consider a triangle A B C , in which D is a point on side B C such that B D = 1 3 B C . We need to prove, 9 A D 2 = 7 A B 2 . Draw A P ⊥ B C , so A P bisect the side B C . ⇒ B P = P C = 1 2 B C = 1 2 A B In a ∆ A B C , A B = B C = A C ⇒ B D = 1 3 B C = 1 3 A B According to the Pythagoras theorem, In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. Therefore, in a triangle A B P , ∠ P = 90 ° A B 2 = A P 2 + B P 2 ⇒ A P 2 = A B 2 - B P 2 ⇒ A P 2 = A B 2 - 1 2 A B 2 = 3 4 A B 2 . . . . . . . . . i Now, applying Pythagoras theorem to the ∆ A D P , ⇒ A D 2 = A P 2 + D P 2 ⇒ A D 2 = 3 4 A B 2 + B P - B D 2 ⇒ A D 2 = 3 4 A B 2 + 1 2 A B - 1 3 A B 2 ⇒ A D 2 = 3 4 A B 2 + 1 36 A B 2 ⇒ 36 A D 2 = 27 A B 2 + A B 2 ⇒ 36 A D 2 = 28 A B 2 Hence, proved.

In the figure, Q T P R = Q R Q S and ∠ 1 = ∠ 2 , prove that ∆ P Q S ~ ∆ T Q R .

Given, Q T P R = Q R Q S . . . . . . . . . . . i and ∠ 1 = ∠ 2 . In ∆ P Q R , we have ∠ 1 = ∠ 2 . ⇒ P R = P Q . . . . . . . . . . . ii ( Sides opposite to equal angles are equal ) From equations i and ii , we can write, Q T P Q = Q R Q S Now, ∠ Q is common angle to ∆ T Q S and ∆ P Q S . Hence, using SAS similarity criterion, we conclude that ∆ P Q S ~ ∆ T Q R .

E M B I B E

Mathematics Class 10>Chapter 8 - Similar Triangles>EXERCISE - 8.4>Q 6

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10th Andhra Pradesh Board

IMPORTANT

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$A B C$ is an isosceles triangle right angle at $C$ . Prove that $A B^{2} = 2 A C^{2}$ .

Important Questions on Similar Triangles

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10th Andhra Pradesh Board

IMPORTANT

Mathematics Class 10>Chapter 8 - Similar Triangles>EXERCISE - 8.4>Q 7

In figure, $O$ is any point in the interior of $∆ A B C$ . If $O D ⊥ B C, O E ⊥ A C$ and $O F ⊥ A B$ , show that

$O A^{2} + O B^{2} + O C^{2} - O D^{2} - O E^{2} - O F^{2} = A F^{2} + B D^{2} + C E^{2}$ .

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10th Andhra Pradesh Board

IMPORTANT

Mathematics Class 10>Chapter 8 - Similar Triangles>EXERCISE - 8.4>Q 7

In figure, $O$ is any point in the interior of $∆ A B C$ . If $O D ⊥ B C, O E ⊥ A C$ and $O F ⊥ A B$ , show that $A F^{2} + B D^{2} + C E^{2} = A E^{2} + C D^{2} + B F^{2}$ .

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Mathematics Class 10>Chapter 8 - Similar Triangles>EXERCISE - 8.4>Q 8

A wire attached to 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?

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Mathematics Class 10>Chapter 8 - Similar Triangles>EXERCISE - 8.4>Q 9

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 in meters.

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10th Andhra Pradesh Board

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Mathematics Class 10>Chapter 8 - Similar Triangles>EXERCISE - 8.4>Q 10

In an equilateral triangle

A B C

,

D

is a point on side

B C

such that

B D = \frac{1}{3} B C

. Prove that

9 A D^{2} = 7 A B^{2}

.

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10th Andhra Pradesh Board

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Mathematics Class 10>Chapter 8 - Similar Triangles>EXERCISE - 8.4>Q 11

In the given figure, $A B C$ is a triangle right-angled at $B$ . $D$ and $E$ are points on $B C$ trisect it. Prove that $8 A E^{2} = 3 A C^{2} + 5 A D^{2}$ .

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10th Andhra Pradesh Board

IMPORTANT

Mathematics Class 10>Chapter 8 - Similar Triangles>EXERCISE - 8.4>Q 12

$A B C$ is an isosceles triangle right-angled at $B$ . Equilateral triangles $A C D$ and $A B E$ are constructed on sides $A C$ and $A B$ . Find the ratio between the areas of $∆ A B E$ and $∆ A C D$ .