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Question
In the adjoining figure, OPQR is a square. A circle drawn with centre O cuts the square in X and Y. Prove that QX = QY.
In the adjoining figure, OPQR is a square. A circle drawn with centre O cuts the square in X and Y. Prove that QX = QY.
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Solution
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 Δ OXP and Δ OYR , we have:
∠ OPX = ∠ ORY
(90° each)
OX = OY
(Radii of a circle)
OP = OR
(Sides of a square)
∴ Δ OXP ≅ Δ OYR (BY RHS congruency rule)
⇒ PX = RY
(By CPCT)
⇒ PQ - PX = QR - RY (PQ and QR are sides of a square)
⇒ QX = QY
Hence, proved.
To prove: QX = QY
Construction: Join OX and OY.
Proof:
In Δ OXP and Δ OYR , we have:
∠ OPX = ∠ ORY
(90° each)
OX = OY
(Radii of a circle)
OP = OR
(Sides of a square)
∴ Δ OXP ≅ Δ OYR (BY RHS congruency rule)
⇒ PX = RY
(By CPCT)
⇒ PQ - PX = QR - RY (PQ and QR are sides of a square)
⇒ QX = QY
Hence, proved.
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