Question

In ΔABC, ∠A = 90° and BC² = 2AC², prove that : ΔABC is isosceles.

Answer 1

Given: ∠A = 90°, BC² = 2AC²

To Prove: ΔABC is isosceles i.e., AB = AC

Proof [Step-wise]:

1. ∠A = 90°

⇒ BC is the hypotenuse of right triangle ΔABC.

2. By the Pythagorean theorem,

BC² = AB² + AC²

3. Substitute the given BC² = 2AC² into step 2:

2AC² = AB² + AC²

4. Rearranging gives AB² = AC².

5. Taking positive square roots side lengths are positive yields AB = AC.

6. Hence, the two sides AB and AC are equal. 

So, ΔABC is isosceles.

ΔABC is isosceles AB = AC.