Multiple choice

Chords $MN$ and $RS$ of a circle intersect at $P$ outside the circle If $PN=3\ cm, MN =5\ cm, PR=2\ cm$ then the value of $SR$ is equal to

  1. $5\ cm$
  2. $8\ cm$
  3. $15\ cm$
  4. $10\ cm$
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D Correct answer
Explanation

By the power of a point theorem, for chords intersecting at P outside the circle, PM * PN = PR * PS. Given PN=3, MN=5, so PM=8. PR=2. Thus 8 * 3 = 2 * PS, so PS = 12. Since PS = PR + RS, 12 = 2 + RS, so RS = 10.

AI explanation

When two secants are drawn from an external point, the Secant Theorem states that the product of the external segment and the whole secant is equal for both lines. For the first secant, the external part is PN equal to 3 cm and the whole length is PM equal to PN plus MN equal to 8 cm, so the product is 3 times 8 equal to 24. For the second secant, the external part is PR equal to 2 cm and the whole length is PS, so 2 times PS equals 24, making PS equal to 12 cm; subtracting PR from PS gives SR equal to 10 cm.

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