Inscribed Squares in Right-Angled Triangle/Side Lengths/Shared Right Angle
Jump to navigation
Jump to search
Theorem
Let $a, b, c$ be the side lengths of a right-angled triangle, where $c$ is the length of the hypotenuse.
Then the side length $l$ of the inscribed square that shares a right angle with the right-angled triangle is given by:
- $l = \dfrac {a b} {a + b}$
Proof
In the figure above, let $BC = a$ and $AC = b$.
Note that $DE \parallel CF$.
Therefore $\triangle BDE \sim \triangle BCA$ by Equiangular Triangles are Similar.
Thus:
\(\ds \frac {BD} {DE}\) | \(=\) | \(\ds \frac {BC} {CA}\) | Definition of Similar Triangles | |||||||||||
\(\ds \frac {a - l} l\) | \(=\) | \(\ds \frac a b\) | ||||||||||||
\(\ds b \paren {a - l}\) | \(=\) | \(\ds a l\) | ||||||||||||
\(\ds b a\) | \(=\) | \(\ds a l + b l\) | ||||||||||||
\(\ds l\) | \(=\) | \(\ds \frac {a b} {a + b}\) |
$\blacksquare$