Question

In the give figure, OPQR is square. A circle drawn with centre O cuts the square in X and Y. Prove that OX = QY.

Answer 1

Given: OPQR is a square. A circle with centre O cuts the square at X and Y.
To prove: QX = QY
Construction: Join OX and OY.

Proof:
In ΔOXP and ΔOYR, we have:
∠OPX = ∠ORY (90° each)
OX = OY (Radii of a circle)
OP = OR (Sides of a square)
∴ ΔOXP ≅ ΔOYR (By RHS congruency rule)
⇒ PX = RY (by CPCT)
⇒ PQ − PX = QR − RY (∵ PQ = QR, sides of the given square)
Hence, QX = QY