Multiple choice

In the circle with centre O as shown chord AB and CD intersect at P and are perpendicular to each other. If AP = $4$ cm, PB = $6$cm and PC = $2$ cm, then the area of the circle is

  1. $\displaystyle 45\pi$
  2. $\displaystyle 49\pi$
  3. $\displaystyle 50\pi$
  4. $\displaystyle 41\pi$
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C Correct answer
AI explanation

By the intersecting chords theorem, the products of the two segments of each chord are equal, so AP * PB = PC * PD. Substituting the known lengths gives 4 * 6 = 2 * PD, which makes PD = 12 cm. This makes the length of chord CD equal to 2 + 12 = 14 cm, and placing the intersection at the origin reveals the circle's endpoints at (-6, 0), (4, 0), (0, 2), and (0, -14); using the equation of a circle, the radius squared is found by solving 6^2 + 7^2 = r^2 + 5^2, yielding r^2 = 50. The area of the circle is pi * r^2, which equals 50 * pi.

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