D and E are points on the sides C A and C B respectively of a triangle A B C right-angled at C . Prove that A E 2 + B D 2 = A B 2 + D E 2 .

In △ A C E , A C 2 + C E 2 = A E 2 … i In △ B C D , B C 2 + C D 2 = B D 2 … i i Adding i and ( i i ) we get: A C 2 + C E 2 + B C 2 + C D 2 = A E 2 + B D 2 … ( i i i ) ⇒ C D 2 + C E 2 + A C 2 + B C 2 = A E 2 + B D 2 In △ C D E , D E 2 = C D 2 + C E 2 In △ A B C , A B 2 = A C 2 + C B 2 Adding both equations and comparing with equation ( i i i ) , we get: D E 2 + A B 2 = A E 2 + B D 2

In a triangle, if square of one side is equal to the sum of the squares of the other two sides then the angle opposite the first side is a right angle. Prove it.

Given, in △ A B C , A C 2 = A B 2 + B C 2 . Construction: Construct a triangle P Q R right-angled at Q such that, P Q = A B and Q R = B C . To prove: ∠ A B C = 90 ° Proof: Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides. In △ P Q R , P R 2 = P Q 2 + Q R 2 ⇒ P R 2 = A B 2 + B C 2 [Given] ⇒ P R 2 = A C 2 [Given] ⇒ P R = A C _____________ 1 In △ A B C and △ P Q R , A B = P Q [Given] B C = Q R [Given] A C = P R [From 1 ] ∴ △ A B C ≅ △ P Q R (By SSS congruency) Corresponding parts of congruent triangles are equal. ⇒ ∠ A B C = ∠ P Q R ⇒ ∠ A B C = 90 ° ∵ ∠ P Q R = 90 ° Hence proved.

E M B I B E

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Triangles>Punjab Board-2019>Q 3

EASY

10th Punjab Board

IMPORTANT

Earn 100

The hypotenuse is the side in right triangle.

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Important Questions on Triangles

MEDIUM

10th Punjab Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Triangles>Punjab Board-2019>Q 4

P Q R

is a triangle right-angled at

P

and

M

is a point on

Q R

such that

P M ⊥ Q R

. Show that

P M^{2} = Q M . M R

.

MEDIUM

10th Punjab Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Triangles>Punjab Board-2019>Q 5

Prove that sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.

HARD

10th Punjab Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 6 - Triangles>Punjab Board-2019>Q 6

If a line is drawn parallel to one side of a triangle to intersects the other two sides in distinct points, the other two sides are divided in the same ratio. Prove it.

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