# What are the basic properties of a 45-45-90 triangle?

Apr 21, 2018

Consider the properties of the sides, the angles and the symmetry.

#### Explanation:

$45 - 45 - 90 \text{ }$ refers to the angles of the triangle.

The color(blue)("sum of the angles is " 180°)

There are $\textcolor{b l u e}{\text{two equal angles}}$, so this is an isosceles triangle.

It therefore also has $\textcolor{b l u e}{\text{ two equal sides.}}$

The third angle is 90°. It is a $\textcolor{b l u e}{\text{right-angled triangle}}$ therefore Pythagoras' Theorem can be used.

The $\textcolor{b l u e}{\text{sides are in the ratio } 1 : 1 : \sqrt{2}}$

It has $\textcolor{b l u e}{\text{one line of symmetry}}$ - the perpendicular bisector of the base (the hypotenuse) passes through the vertex, (the 90° angle).

It has $\textcolor{b l u e}{\text{no rotational symmetry.}}$