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.
