Chiu Chang Suann Jing/Examples/Example 9

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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 $15$?


The diameter of the circle is $6$.


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:

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