Question

In ◻ABCD, ∠B = 90° and ∠D = 90°. Prove that : 2AC² = AB² + BC² + CD² + DA².

Answer 1

Given: In quadrilateral ABCD, ∠B = 90° and ∠D = 90°.

To Prove: 2AC² = AB² + BC² + CD² + DA²

Proof (Step-wise):

1. Consider triangle ABC.

Since ∠B = 90°, triangle ABC is right-angled at B.

By the Pythagorean theorem,

AC² = AB² + BC²

2. Consider triangle ADC.

Since ∠D = 90°, triangle ADC is right-angled at D.

By the Pythagorean theorem,

AC² = AD² + DC²

3. Add the two equations from steps 1 and 2:

AC² + AC² = (AB² + BC²) + (AD² + DC²)

4. Simplify:

2AC² = AB² + BC² + CD² + DA²

Hence proved.