Carcass wrote:
In the figure above, the five circles have points in common as shown. P is the center of the largest circle, Q and R are centers of the medium-sized circles, and Q, P, and R are points on a straight line. What fraction of the largest circular region is shaded?
(A) \(\frac{1}{16}\)
(B) \(\frac{1}{8}\)
(C) \(\frac{3}{16}\)
(D) \(\frac{1}{4}\)
(E) \(\frac{1}{2}\)
Let's start by assigning a "nice" value to the radius of the smallest circle.
Let's say the
blue circle has radius
1If the
blue circle has radius
1, then the DIAMETER of the
blue circle must be 2.
This means the radius of the
green circle must be
2If the radius of the
green circle is
2, then the DIAMETER of the
green circle must be 4.
This means the radius of the
red circle must be
4Area of circle \(= \pi r^2\)So, the area of the
red circle \(= \pi (4^2) = 16\pi\)
And the area of ONE
blue circle \(= \pi (1^2) = \pi\)
So, the area of TWO
blue circles \(= 2\pi\)
What fraction of the largest circular region is shaded?Answer \(= \frac{2\pi}{16\pi} = \frac{1}{8}\)
Answer: B