# Right-Angled Triangle A **right-angled triangle** is a [[triangle]] in which one of the angles is a [[right angle]]. The side opposite the right-angle is known as the **hypotenuse** of the triangle. ## Calculating Lengths In a right-angled triangle the lengths of the sides are related via [[Pythagoras Theorem]]: with side lengths $a$, $b$, and $c$ where $c$ is the hypotenuse then the following identity holds: $$ a^2 + b^2 = c^2 $$ ## Division into Isosceles Triangles A useful fact about right-angled triangles is that they can be divided into two [[isosceles triangles]]. Mark a point at the [[midpoint]] of the hypotenuse and join it to the [[vertex]] where the right-angle is located. This splits the triangle into two isosceles triangles. One way to see this is to complete the triangle to a [[rectangle]]. The hypotenuse becomes a diagonal of the rectangle. The other diagonal crosses this hypotenuse at its midpoint, and [[symmetry]] shows that the four half-diagonals are all the same length. In the rectangle there are four isosceles triangles and two of them coincide with the original right-angled triangle. [[!redirects hypotenuse]] [[!redirects right-angled triangles]] [[!redirects Right-Angled Triangle]] [[!redirects Right-angled triangle]] category: shapes category: triangles