Multiple choice

PQ and QR are the two chords of a circle. Line segment RP is stretched to a point S and then SQ is joined which intersects the circle at T. PQ = QR = 9 cm, QT = 4 cm. What is the measure of SQ?

  1. 18 cm

  2. 20 cm

  3. 21.25 cm

  4. 20.25 cm

  5. a
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D Correct answer
Explanation

By the Power of a Point theorem for the circle, if RP is extended to S, then SP * SR = ST * SQ. Let SQ = x. Then ST = x - 4. Since PQ = QR = 9, the triangle is isosceles. Using geometry properties, the segment lengths satisfy the power of a point relation, resulting in x = 20.25.

AI explanation

Since PQ equals QR, triangle PQR is isosceles, and the angle QPR equals angle QRP. By the alternate segment theorem or equal inscribed angles, angle QST equals angle QPS. This makes triangle QPT similar to triangle QSP, so the ratio of corresponding sides gives QP divided by QS equals QT divided by QP. Substituting the values gives 9 divided by QS equals 4 divided by 9. Solving for QS yields QS equals 81 divided by 4, which is 20.25 cm.

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