# Nested Semi-Circles +-- {.image} [[NestedSemiCircles.png:pic]] > What fraction of the largest semicircle is shaded? =-- ## Solution by [[Chord Properties]] and [[Pythagoras' Theorem]] +-- {.image} [[NestedSemiCirclesLabelled.png:pic]] =-- In the above diagram, $O$ is the [[midpoint]] of $A B$ and so is the centre of the smallest semi-circle while $Q$ is the [[midpoint]] of $C D$ and so is the centre of the middle semi-circle. As $A B$ is a chord of the middle semi-circle, the [[perpendicular bisector]] of $A B$ passes through its centre. That bisector passes through the midpoint, which is $O$, and so $O Q$ is perpendicular to $A B$. The line segment $C D$ is tangential to the smallest circle and so is perpendicular to $O Q$ since the [[angle between a radius and tangent]] is a [[right angle]]. This means that $O Q$ is also the [[perpendicular bisector]] of $C D$ and so $O$ is also the centre of the largest semi-circle. Let $a$, $b$, $c$ be the radii of the three semi-circles in increasing order. Applying [[Pythagoras' theorem]] to triangle $Q O A$ shows that $b^2 = a^2 + a^2 = 2 a^2$. Applying it to triangle $O Q C$ shows that $c^2 = b^2 + a^2 = 3 a^2$. The area of the largest semi-circle is therefore $3$ times that of the smallest, so $\frac{1}{3}$ of the largest is shaded.