If ΔABC is an isosceles triangle such that ∠ABC = 90°, then the true statement…

Concept

In a right triangle, the Pythagorean theorem states that the square of the hypotenuse equals the sum of the squares of the two legs. In an isosceles right triangle, the two legs are equal.

Application

1. Because ∠ABC = 90°, AC is the hypotenuse, and AB and BC are the legs.

2. Since ΔABC is isosceles with equal legs at the right angle, AB = BC.

3. By the Pythagorean theorem, (AC)2 = (AB)2 + (BC)2.

4. Substitute BC = AB: (AC)2 = (AB)2 + (AB)2 = 2(AB)2.

Cross-check

Let AB = BC = a. Then AC = a√2, so (AC)2 = 2a2 = 2(AB)2, confirming the relationship.