Properties of a Right-Angled Triangle

A right-angled triangle has one angle of $90^ ext{o}$. The side opposite this angle is the hypotenuse. The triangle satisfies the Pythagorean Theorem: $a^2 + b^2 = c^2$, where $a$ and $b$ are the legs and $c$ is the hypotenuse. The triangle has special properties related to its medians, altitudes (heights), and circles (incircle and circumcircle).

Important properties

• The altitude from the right angle to the hypotenuse divides the triangle into two smaller right-angled triangles, each similar to the original.

• The length of the median from the right angle to the hypotenuse is half the hypotenuse.

• The area is $\frac{1}{2}ab$ where $a$ and $b$ are the legs.

• The radius $r$ of the incircle is $r = \frac{a + b - c}{2}$.

• The circumcircle's center is at the midpoint of the hypotenuse, and its radius is $\frac{c}{2}$.