# GEOM 2 | Lesson 3 |Making Connections (Special Right Triangles) ### 45-45-90 Triangles

From Try This! Special Right Triangles #1, the relationship between the sides of two 45-45-90 triangles showed that the hypotenuse was equal to the length of the leg multiplied by $$\sqrt{2}$$.

The proof below shows that this relationship is true for any 45-45-90 triangle with leg length of $$x$$. Use the Pythagorean Theorem to find the length of the hypotenuse.

• $$x^2+x^2=y^2$$
• $$2x^2=y^2$$
• $$\sqrt{x^2*2}=y^2$$
• $$x\sqrt{2}=y$$

Thus, $$y=x\sqrt{2}$$.

### 30-60-90 Triangles

From Try This! Special Right Triangles #2, the relationship between the sides of two 30-60-90 triangles showed that the hypotenuse was equal twice the length of the the shortest leg, and the length of the longer leg was $$\sqrt{3}$$ times the length of the shortest leg.

The proof below shows that this relationship is true for any 30-60-90 triangle with shortest leg length of $$x$$. First, since the 30-60-90 triangle is formed by bisecting one angle of an equilateral triangle, the length of the longest side is twice the shortest leg, or $$2x$$.

Second, use the Pythagorean Theorem to find the length of the largest leg.

• $$x^2+y^2=(2x)^2$$
• $$4x^2-x^2=y^2$$
• $$y^2=3x^2$$
• $$y=x\sqrt{3}$$

Thus, the hypotenuse = $$2x$$ and the longest side = $$x\sqrt{3}$$.

If you forget these formulas, you can always construct the square or an equilateral triangle and divide it into two 45-45-90  or 3-60-90 triangles, label the congruent parts, and use the Pythagorean Theorem to find the remaining missing parts.