Question

In a square PQRS, diagonals bisect each other at O. Prove that ΔPOQ ≅ ΔQOR ≅ ΔROS ≅ ΔSOP.

Answer 1

Given: PQ = QR = RS = PS and SO = QO = PO = RO

To Prove: ΔPOQ ≅ ΔQOR ≅ ΔROS ≅ ΔSOP

Proof:

In ΔSOR and ΔPOQ:

SR = PQ (Given)

RO = PO (Given)

SO = OQ (Given)

∴ ΔSOR ≅ ΔPOQ … (1) (SSS congruency)

In ΔPOS and ΔROQ:

PS = QR (Given)

PO = OR (Given)

SO = OQ (Given)

∴ ΔSOP ≅ ΔROQ … (2) (SSS congruency)

In ΔPOS and ΔROS:

PS = SR (Given)

OS = OS (Common)

PO = OR (Given)

∴ ΔPOS ≅ ROS … (3) (SSS congruency)

In ΔPOQ and ΔROQ:

PQ = QR (Given)

OQ = OQ (Common)

PO = OR (Given)

∴ ΔPOQ ≅ ROQ … (4) (SSS congruency)

From (1), (2), (3) and (4):

ΔPOQ ≅ ΔQOR ≅ ΔROS ≅ ΔSOP