Question

In a quadrilateral PQRS, ∠Q = ∠S = 90° then prove that 2PR² – QR² = PQ² + PS² + SR².

Answer 1

Given:

A quadrilateral PQRS where angle Q is 90 degrees and angle S is 90 degrees.

Now,

∴ PR² = PQ² + QR²  ...[Applying the Pythagorean theorem in right-angled triangle PQR]

∴ PR² = PS² + SR²  ...[Applying the Pythagorean theorem in right-angled triangle PSR]

∴ PR² + PR² = (PQ² + QR²) + (PS² + SR²)  ...[Adding both equations together]

∴ 2PR² = PQ² + QR² + PS² + SR²  ...[Combining like terms on the left side]

∴ 2PR² − QR² = PQ² + PS² + SR²  ...[Rearranging terms by subtracting the square of QR from both sides]

∴ 2PR² − QR² = PQ² + PS² + SR²