# Chiu Chang Suann Jing/Examples/Example 9

## Example of Problem from Chiu Chang Suann Jing

What is the largest circle that can be inscribed within a right-angled triangle,
the two short sides of which are respectively $8$ and $15$?

## Solution

The diameter of the circle is $6$.

## Proof

Let the lengths of the short sides be denoted $a$ and $b$.

Let the length of the hypotenuse be denoted $c$.

Without loss of generality let $a \le b$.

Let the right-angled triangle be half of a rectangle whose sides are of length $a$ and $b$.

Let the rectangle be dissected along the straight lines shown.

Let $2$ copies of this rectangle be assembled into one long rectangle whose length is $a + b + c$ and whose width is $D$.

Note that $D$ is the diameter of the circle that we are required to find. Thus we have that:

$D \paren {a + b + c} = 2 a b$

Setting $a = 8$, $b = 15$, from Pythagoras's Theorem we have that $c = 17$.

Thus we have that:

 $\ds D$ $=$ $\ds \dfrac {2 a b} {a + b + c}$ $\ds$ $=$ $\ds \dfrac {2 \times 8 \times 15} {8 + 15 + 17}$ $\ds$ $=$ $\ds \dfrac {240} {40}$ $\ds$ $=$ $\ds 6$

$\blacksquare$