Multiple choice

P is a point outside the circle at a distance of 6.5 cm from the centre O. PR is a secant, such that it intersects the circle at Q and R. If PQ = 4.5 cm and QR = 3.5 cm, then what is the radius (in cm) of the circle?

  1. 1.5

  2. 2

  3. 2.5

  4. 3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Power of point P = PQ * PR = 4.5 * (4.5 + 3.5) = 4.5 * 8 = 36. Also, Power = d^2 - r^2, where d=6.5. 36 = 6.5^2 - r^2 = 42.25 - r^2. r^2 = 42.25 - 36 = 6.25. r = 2.5.

AI explanation

The entire length of the secant from point P is PR = PQ + QR = 4.5 + 3.5 = 8 cm. By the tangent-secant theorem (or power of a point), the square of the tangent length equals the product of the external secant length and the total secant length, but since no tangent is given, we instead drop a perpendicular from centre O to chord PR. Using the property of intersecting chords (Power of a Point), OP^2 equals the square of the radius plus the product of the secant segments, so 6.5^2 = r^2 + (4.5 * 8), yielding 42.25 = r^2 + 36. Therefore, the radius squared is 6.25 and the radius is 2.5 cm.

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