Multiple choice

Two concentric circles are such that the smaller divides the larger circle into two regions of equal areas. If the radius of the smaller circle is $2$, then the length of a tangent from any point p on the larger circle to the smaller circle is

  1. $\sqrt{2}$
  2. $2$
  3. $1$
  4. $2$ $\sqrt{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let R be radius of larger circle, r=2 be radius of smaller. Area of smaller = pi*r^2 = 4pi. Area of larger = 2 * 4pi = 8pi. So pi*R^2 = 8pi, R^2 = 8, R = 2*sqrt(2). A tangent from a point on the larger circle to the smaller circle forms a right triangle with the radius of the smaller circle and the distance from center to point (R). Length^2 = R^2 - r^2 = 8 - 4 = 4. Length = 2.

AI explanation

Because the smaller circle divides the larger circle into two regions of equal areas, the area of the smaller circle is exactly half the area of the larger circle, making the radius of the larger circle 2 times the square root of 2. The length of a tangent drawn from an external point to a circle is found using the Pythagorean theorem by taking the square root of the difference of the squares of the radii. Calculating the square root of 8 minus 4 gives a tangent length of 2.

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