### Areas and Ratios

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

### Six Discs

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

Using Pythagoras` theorem in $ABC$,
$$\begin{eqnarray}a &=& \sqrt{b^2 + c^2} \\ \mbox{Area of semicircle on BC} &=& \pi \frac {a^2}{4}\\ &=& S1\\ \mbox{Area of semicircle on AC} &=& \pi \frac {b^2}{4}\\ &=& S2 \\ \mbox{Area of semicircle on AB} &=& \pi \frac{c^2}{4}\\ &=& S3\\ \mbox{Area of crescents} &=& S2 + S3 + \mbox{Area ABC} - S1\\ &=& \pi \frac {b^2}{4} +\pi \frac {c^2}{4} + \mbox{Area ABC} - \pi \frac {a^2}{4}\\ &=& \pi \frac {b^2}{4} + \pi \frac{c^2}{4} - \pi \frac {a^2}{4} + \mbox{Area ABC}\\ &=& \frac{\pi}{4} \times ( b^2 + c^2 - a^2 ) + \mbox{Area ABC}\\ &=& 0 + \mbox{Area ABC} \\ &=& \mbox{Area ABC} \end{eqnarray}$$