Two circles touch externally at P QR is a common tangent of the circles touching the circles at Q an...........

Quantitative Aptitude ( CCS) Geometry

Two circles touch externally at P, QR is a common tangent of the circles touching the circles at Q and R. Then measure of ∠QPR is

90°

60°

45°

30°

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Explanation:

∠POQ = ∠POR = 90°
OQ = OP = OR [? Tangent drawn from the same external point]
∴ ∠OQP = ∠OPQ = ∠ORP = ∠OPR
In ΔPOQ, we know that
∠POQ + ∠OQP + ∠OPQ = 180°
90° + ∠OPQ + ∠OPQ = 180°
2∠OPQ = 180° – 90° = 90°
∠OPQ = 45°
Similarly in ΔPOR, we get
∠ORP = 45°
∴ ∠QPR = ∠OPQ + ∠ORP = 45° + 45° = 90°