Arc radius
Two arcs are drawn in a right-angled triangle as shown. What is the length $r$?
Age
14 to 16
Challenge level
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 $$\begin{align}(2r)^2+6^2=&(r+6)^2\\
\Rightarrow4r^2+36=&r^2+12r+36\\
\Rightarrow3r^2=&12r\\
\Rightarrow3r^2-12r=&0\\
\Rightarrow3r(r-4)=&0\\
\Rightarrow r-4=&0\hspace{12mm}(r\ne0)\\
\Rightarrow r=&4\end{align}$$