Multiple choice

The centre of a circle is at $O$. $AB$ and $CD$ are two chords of length $d$ and $l$ respectively. If $P$ is the mid point of $CD$, then the length $OP$ is

  1. $\sqrt{d^2+l^2}$
  2. $\sqrt{d^2-l^2}$
  3. $\displaystyle\frac{1}{2}\sqrt{d^2+l^2}$
  4. $\displaystyle\frac{1}{2}\sqrt{d^2-l^2}$
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D Correct answer
AI explanation

Assuming d is the diameter, the radius squared is d^2 / 4. Let OP be the perpendicular distance from the center to the chord CD. By the Pythagorean theorem, OP^2 + (l/2)^2 equals the radius squared. Substituting the radius gives OP^2 + l^2/4 = d^2/4, which leads to OP = 1/2 * sqrt(d^2 - l^2). The length is 1/2 sqrt(d^2 - l^2).

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