Multiple choice

Two circles whose radii are equal to 4 and 8 intersect at right angles. The length of their common chord is

  1. $\displaystyle \frac{16}{\sqrt{5}}$
  2. $8$
  3. $\displaystyle 4\sqrt{6}$
  4. $\displaystyle \frac{8\sqrt{5}}{5}$
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A Correct answer
Explanation

For two intersecting circles with radii r1, r2 and distance d between centers, if they intersect at right angles, d^2 = r1^2 + r2^2. Here d^2 = 4^2 + 8^2 = 16 + 64 = 80, so d = sqrt(80) = 4*sqrt(5). The length of the common chord is (2 * r1 * r2) / d = (2 * 4 * 8) / (4 * sqrt(5)) = 16 / sqrt(5).

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