# Special Right Triangles

## Key Questions

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

• $m a t h b f \left\{{30}^{\circ} \text{-"60^circ"-} {90}^{\circ}\right\}$ Triangle

The ratios of three sides of a ${30}^{\circ} \text{-"60^circ"-} {90}^{\circ}$ triangle are:

$1 : \sqrt{3} : 2$

I hope that this was helpful.

• Each black-and-red (or black-and-yellow) triangles is a special right-angled triangle. The figures outside the circle - $\frac{\pi}{6} , \frac{\pi}{4} , \frac{\pi}{3}$ - are the angles that the triangles make with the horizontal (x) axis. The other figures - $\frac{1}{2} , \frac{\sqrt{2}}{2} , \frac{\sqrt{3}}{2}$ - are the distances along the axes - and the answers to $\sin \left(x\right)$ (yellow) and $\cos \left(x\right)$ (red) for each angle.

• Special Right Triangles

1. ${30}^{\circ}$-${60}^{\circ}$-${90}^{\circ}$ Triangles whose sides have the ratio $1 : \sqrt{3} : 2$

2. ${45}^{\circ}$-${45}^{\circ}$-${90}^{\circ}$ Triangles whose sides have the ratio $1 : 1 : \sqrt{2}$

These are useful since they allow us to find the values of trigonometric functions of multiples of ${30}^{\circ}$ and ${45}^{\circ}$.