A B C is an isosceles triangle right angled at C . Prove that A B 2 = 2 A C 2 .

Given, A B C is an isosceles triangle right angle at C . To prove: A B 2 = 2 A C 2 Proof: An isosceles right triangle is a right triangle that consists of two equal length legs. B C and A C are the legs of the given isosceles right triangle A B C . Thus, B C = A C . . . ( i ) 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 right △ A B C , A B 2 = A C 2 + B C 2 ⇒ A B 2 = A C 2 + A C 2 ∵ B C = A C ⇒ A B 2 = 2 A C 2 Hence, proved.

E M B I B E

Understanding ICSE Mathematics Class 9>Chapter 12 - Pythagoras Theorem>EXERCISE 12>Q 17

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IMPORTANT

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In $△ A B C, A B = A C = x cm, B C = 10 cm$ and the area of $△ A B C$ is $60 {cm}^{2}$ . Find $x$ .

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Important Questions on Pythagoras Theorem

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Understanding ICSE Mathematics Class 9>Chapter 12 - Pythagoras Theorem>EXERCISE 12>Q 18

In a rhombus, if diagonals are

30 cm and 40 cm

, find its perimeter (in

cm

).

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Understanding ICSE Mathematics Class 9>Chapter 12 - Pythagoras Theorem>EXERCISE 12>Q 19

In the figure given below, $A B | | D C, B C = A D = 13 cm, A B = 22 cm and D C = 12 cm .$ Calculate the height (in $cm$ ) of the trapezium $A B C D$ .

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Understanding ICSE Mathematics Class 9>Chapter 12 - Pythagoras Theorem>EXERCISE 12>Q 19

In the figure given below, $A B ∥ D C, ∠ A = 90 °, D C = 7 cm, A B = 17 cm a n d A C = 25 cm .$ Calculate $B C$ (in $cm$ ).

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Understanding ICSE Mathematics Class 9>Chapter 12 - Pythagoras Theorem>EXERCISE 12>Q 20

$AD$ is perpendicular to the side $BC$ of an equilateral $△ ABC$ . Prove that $4 {AD}^{2} = 3 {AB}^{2}$ .

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Understanding ICSE Mathematics Class 9>Chapter 12 - Pythagoras Theorem>EXERCISE 12>Q 22

If

A D, B E

and

C F

are medians of

△ A B C

, prove that

3 (A B^{2} + B C^{2} + C A^{2}) = 4 (A D^{2} + B E^{2} + C F^{2}) .

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Understanding ICSE Mathematics Class 9>Chapter 12 - Pythagoras Theorem>EXERCISE 12>Q 23

In the figure, the diagonals $A C$ and $B D$ of a quadrilateral $A B C D$ intersect at $O$ , at the right angles. Prove that $A B^{2} + C D^{2} = A D^{2} + B C^{2} .$