Chords in Circle and Length Calculation

CAT 2019 Slot 1 · QA · Medium · Geometry

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AB is a diameter of a circle of radius 5 cm. Let P and Q be two points on the circle so that the length of PB is 6 cm, and the length of AP is twice that of AQ. Then the length, in cm, of QB is nearest to

  1. A.

    8.5

  2. B.

    9.3

  3. C.

    9.1

  4. D.

    7.8

Answer

C

Explanation

AB=10 cmAB = 10\text{ cm}. Since ∠APB=90∘\angle APB = 90^\circ, AP=102−62=8 cmAP = \sqrt{10^2 - 6^2} = 8\text{ cm}. Given AP=2AQ  ⟹  AQ=4 cmAP = 2 AQ \implies AQ = 4\text{ cm}. In right △AQB\triangle AQB (as ∠AQB=90∘\angle AQB = 90^\circ), QB=102−42=84≈9.165≈9.1 cmQB = \sqrt{10^2 - 4^2} = \sqrt{84} \approx 9.165 \approx 9.1\text{ cm}.

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