Special right triangles

A special right triangle is a right triangle whose angles and ratios of sides make the triangle useful.

45-45-90 Triangles

Main article: 45-45-90 triangle

A 45-45-90 triangle is a right triangle that has two $45^\circ$ angles. All 45-45-90 triangles are isosceles, so let's call both legs of the triangle $x$ . If that is the case, then the hypotenuse will always be $x\sqrt2$ .

30-60-90 Triangles

Main article: 30-60-90 triangle

A 30-60-90 triangle is a right triangle that has a $30^\circ$ angle and a $60^\circ$ angle. Let's call the side opposite of the $30^\circ$ angle $x$ . Then, the side opposite of the $60^\circ$ angle would have a length of $x\sqrt 3$ . Finally, the hypotenuse of a 30-60-90 Triangle would have a length of $2x$ . There is also the ratio of $1:\sqrt3:2$ . With 2 as the hypotenuse and 1 opposite of the 30 degrees. That leaves $\sqrt3$ as the only length left.

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Special right triangles

45-45-90 Triangles

30-60-90 Triangles

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