# Two Special Triangles

You should know the relationship between the lengths of the sides in two special triangles:

- $\,30^\circ{-}60^\circ{-}90^\circ\,$ triangles
- $\,45^\circ{-}45^\circ{-}90^\circ\,$ triangles

Conversely, if a triangle has sides of lengths $\,s\,,$ $\,\sqrt{3}s\,,$ and $\,2s\,,$ then it is a $\,30^\circ\text{-}60^\circ\text{-}90^\circ\,$ triangle.

Since $\,\sqrt{3}\approx 1.7\,,$ it follows that the side opposite the $\,60^\circ\,$ angle is a little more than one and a half times the shortest side.

Conversely, if a triangle has sides of lengths $\,s\,,$ $\,s\,,$ and $\,\sqrt{2}s\,,$ then it is a $\,45^\circ\text{-}45^\circ\text{-}90^\circ\,$ triangle.

Since $\,\sqrt{2}\approx 1.4\,,$ it follows that the hypotenuse is a little less than one and a half times the shortest side.

It's easy to see that these are the correct relationships between the lengths of the sides, as follows:

## $\,30^\circ\text{-}60^\circ\text{-}90^\circ\,$ Triangle Relationships

Start with an equiangular (hence equilateral) triangle, where each side has length $\,2\,.$ Notice that all the angles must be $\,\frac{180^\circ}{3} = 60^\circ\,.$

Drop a perpendicular, as shown below. The purple and green triangles are $\,30^\circ\text{-}60^\circ\text{-}90^\circ\,$ triangles, where the length of the shortest side is $\,1\,.$

A quick application of the Pythagorean Theorem shows that the remaining side has length $\,\sqrt{3}\,.$ Then, scale all sides by $\,s\,.$

## $\,45^\circ\text{-}45^\circ\text{-}90^\circ\,$ Triangle Relationships

Create a triangle with two sides of length $\,1\,$ and a $\,90^\circ\,$ angle, as shown below.

Since angles opposite equal sides must be equal, and since the two remaining angles must add to $\,90^\circ\,,$ they must each be $\,45^\circ\,.$

A quick application of the Pythagorean Theorem shows that the remaining side must equal $\,\sqrt{2}\,.$ Then, scale all sides by $\,s\,.$