Length of line segment in cyclic geometry

CAT 2018 Slot 2 · QA · Medium · Geometry

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On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, is

Answer

24

Explanation

Since BC is the diameter of the circle: ∠BPC=90∘\angle BPC = 90^\circ and ∠BQC=90∘\angle BQC = 90^\circ.

Thus, CP⊥ABCP \perp AB and BQ⊥ACBQ \perp AC. CP and BQ are altitudes of △ABC\triangle ABC.

The area of △ABC\triangle ABC can be expressed in two ways: Area=12×AB×CP=12×AC×BQ\text{Area} = \frac{1}{2} \times AB \times CP = \frac{1}{2} \times AC \times BQ 12×30×20=12×25×BQ\frac{1}{2} \times 30 \times 20 = \frac{1}{2} \times 25 \times BQ 600=25×BQ  ⟹  BQ=24 cm600 = 25 \times BQ \implies BQ = 24\text{ cm}

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