# Arc Radius

Two arcs are drawn in a right-angled triangle as shown. What is the length $r$?

## Problem

Triangle ABC is right-angled at A and side AB is 6 cm long.

An arc of radius $r$ cm is drawn with centre C such that it bisects side AC.

An arc of radius 6 cm is drawn with centre B such that the arcs both meet BC at the same point, as shown below.

Image

Find the value of $r$.

*This problem is adapted from the World Mathematics Championships*

## Student Solutions

More lengths are show on the diagrams below.

Applying Pythagoras' theorem to the diagram on the right gives $$\begin{align}(2r)^2+6^2=&(r+6)^2\\

\Rightarrow4r^2+36=&r^2+12r+36\\

\Rightarrow3r^2=&12r\\

\Rightarrow3r^2-12r=&0\\

\Rightarrow3r(r-4)=&0\\

\Rightarrow r-4=&0\hspace{12mm}(r\ne0)\\

\Rightarrow r=&4\end{align}$$

Image

Applying Pythagoras' theorem to the diagram on the right gives $$\begin{align}(2r)^2+6^2=&(r+6)^2\\

\Rightarrow4r^2+36=&r^2+12r+36\\

\Rightarrow3r^2=&12r\\

\Rightarrow3r^2-12r=&0\\

\Rightarrow3r(r-4)=&0\\

\Rightarrow r-4=&0\hspace{12mm}(r\ne0)\\

\Rightarrow r=&4\end{align}$$