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45°-45°-90° Triangles

There is a special relationship among the measures of the sides of a $45°-45°-90°$ triangle.

$45°-45°-90°$ triangle is a commonly encountered right triangle whose sides are in the proportion $1:1:\sqrt{2}$ . The measures of the sides are $x$ , $x$ , and $x\sqrt{2}$ .

To see why this is so, note that by the Converse of the Pythagorean Theorem , these values make the triangle a right triangle.

$\begin{array}{l}{x}^2+x^2=2x^2\\ ={\left(x\sqrt{2}\right)}^2\end{array}$

Note that an isosceles right triangle must be a $45°-45°-90°$ triangle.