Multiple choice

$X$ is a point outside of a circle whose center is $O$. From $X$, a tangent whose length is $a$ is drawn to the circle and the shortest distance from $X$ to the circle is $\dfrac{a}{2}$. Find the radius of the circle.

  1. $\dfrac{3a}{4}$
  2. $\dfrac{3a}{2}$
  3. $\dfrac{1}{2}a$
  4. $a$
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A Correct answer
Explanation

Let R be the radius. Distance from center O to X is d = sqrt(a^2 + R^2). Shortest distance to circle is d - R = a/2. So d = R + a/2. Substituting: (R + a/2)^2 = a^2 + R^2. R^2 + aR + a^2/4 = a^2 + R^2. aR = 3a^2/4. R = 3a/4.

AI explanation

The line connecting the external point to the center of the circle forms the hypotenuse of a right triangle with the radius and the tangent. Since the tangent length is a and the shortest distance from the point to the circle is a divided by 2, the total distance from the external point to the center is a plus a divided by 2, which equals 3a divided by 2. Applying the Pythagorean theorem gives us r squared plus a squared equals the quantity of 3a divided by 2 squared, resulting in r equals 3a divided by 4.

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