Multiple choice

In a circle chords $QP$ and $SR$ intersect externally at A. If $AP = 7 cm, PQ = 14 cm, AR = 8 cm$ then the ratio of $AR : RS$.

  1. $64 : 83$
  2. $83 : 64$
  3. $64 : 81$
  4. $81 : 64$
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A Correct answer
Explanation

By the secant-secant theorem, AQ * AP = AS * AR. Given AP=7, PQ=14, so AQ = 21. AR=8. Let RS=x, so AS = 8+x. 21 * 7 = (8+x) * 8. 147 = 64 + 8x. 8x = 83. x = 83/8. The ratio AR : RS = 8 : (83/8) = 64 : 83.

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