The familiar yin-yang symbol is the interior of a circle divided into two regions by the joining of two half-circles of half the radius of the big circle, as in the figure below. You are to draw a single straight line segment that divides each of the two areas, yin and yang, in half. Explain why your line segment works!

## Solution

If we let the large circle have radius $2$ then each quadrant has area $\pi$ and each half of the small circle has area $\pi /2$. This information is included the figure below, part [A].

It follows that yin and yang are each area $2 \pi$ and a 45° line will divide the figure the way we want. In the figure, part [B], on each side of the line there is $\pi$ worth of yin and ditto of yang.