JAMB 1992 · UME · Q26

Circle geometry

In the figure, $PQR$ is a semicircle, and $PQ$ and $QR$ are chords. $QS$ is the perpendicular from $Q$ to the diameter $PR$ . What is the expression for $QS$ ?

Worked solution (try it first)

$PR$ is a diameter, so $\angle PQR = 90^\circ$ (angle in a semicircle).
$QS$ is perpendicular to $PR$ , so triangles $PSQ$ and $QSR$ are similar and $\dfrac{PS}{QS} = \dfrac{QS}{SR}$ .
Cross-multiply: $QS^2 = PS \cdot SR$ , so $QS = \sqrt{PS \cdot SR}$ , option B.

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