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# Special Triangle Ratios

## Ratios based on 45-45-90 and 30-60-90 triangles.

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Special Triangles

Lengths of Sides in an Isosceles Right Triangle

*Remember that the you can find out if the triangle is a right triangle using the Pythagoream Theorem*

Isosceles Right Triangle: An isosceles right triangle is a triangle with one angle equal to ninety degrees and each of the other two angles equal to forty five degrees.

[Figure1]

a2+a22a22a2a2=c2=c2=c2=c\begin{align*}a^2 + a^2 & = c^2\\ 2a^2 & = c^2\\ \sqrt{2a^2} & = \sqrt{c^2}\\ a\sqrt{2} & = c\end{align*}

Relationship of Sides in a 30-60-90 Triangle

30-60-90 Triangle: A 30-60-90 Triangle: is a triangle with one angle equal to ninety degrees, one angle equal to thirty degrees, and one angle equal to sixty degrees.

[Figure2]

s2+h2s2+h2h2h=(2s)2=4s2=3s2=s3\begin{align*}s^2 + h^2 & = (2s)^2\\ s^2 + h^2 & = 4s^2\\ h^2 & = 3s^2\\ h & = s\sqrt{3}\end{align*}

Special Triangle Ratios

Special Triangle: A special triangle is a triangle that has particular internal angles that cause the sides to have a certain length relationship with each other.

What are some special triangles that can be identified?

The ratio of the sides of an isosceles right triangle is x:x:x2\begin{align*}x:x:x\sqrt{2}\end{align*}

The ratio of the sides of a 30-60-90 right triangle is x:x3:2x\begin{align*}x:x\sqrt{3}:2x\end{align*}

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