Multiple choice

In the circle shown AB = 24 and the perpendicular chord CD bisects AB If DM is 4 times as long as CM then the length of BD is

  1. $\displaystyle 8\sqrt{5}$
  2. $\displaystyle 12\sqrt{5}$
  3. $\displaystyle 16\sqrt{5}$
  4. $\displaystyle 20\sqrt{5}$
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B Correct answer
AI explanation

Since CD is the perpendicular bisector of chord AB at its midpoint M, AM = 12. Let CM = x, so DM = 4x, and the total diameter length CD = 5x; if the centre is O, the radius is 5x/2, and OM = 5x/2 - x = 3x/2. Using the intersecting chords property, AM^2 = CM * DM, so 144 = x * 4x, which means x = 6; therefore, CM = 6 and DM = 24. In right triangle BMD, BD^2 = BM^2 + DM^2 = 144 + 576 = 720, so BD = 12 * sqrt(5).

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