Multiple choice

P is a point outside a circle with centre O, and it is 14 cm away from the centre. A secant PAB drawn from P intersects the circle at the points A and B such that PA = 10 cm and PB = 16 cm. The diameter of the circle is:

  1. 10 cm

  2. 13 cm

  3. 12 cm

  4. 11 cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Power of point P = PA * PB = 10 * 16 = 160. Also, Power = d^2 - r^2, where d = 14. 160 = 14^2 - r^2 = 196 - r^2. r^2 = 36, so r = 6. Diameter = 2r = 12.

AI explanation

Using the secant and tangent length identity, the square of the tangent length from an external point equals the product of the external secant length and the total secant length. We set up the equation PT squared equals PA times PB, which is PT squared equals 10 times 16, giving a tangent length of PT = square root of 160 cm. A right triangle is formed by the tangent, the radius to the tangent point, and the line from the external point to the center, allowing us to use the Pythagorean theorem to find the radius. The radius squared equals PO squared minus PT squared, which becomes 14 squared minus 160, equaling 196 minus 160 = 36. The radius is the square root of 36, which is 6 cm, making the diameter 2 * 6 = 12 cm.

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