Summary of Special Right Triangles

45°-45°-90° Triangle

• Formed by an isosceles right triangle, where both legs are equal.
• Hypotenuse: $c = leg \cdot \sqrt{2}$
• Ratios of sides: $1:1:\sqrt{2}$
• Applications:
  • If leg = 4, then hypotenuse = $4\sqrt{2}$.
  • If hypotenuse = $3\sqrt{2}$, then leg = 3.

30°-60°-90° Triangle

• Based on an equilateral triangle, when cut in half.
• Ratios of sides: $1:2:\sqrt{3}$
• Hypotenuse = $2 \times short \ leg$
• Applications:
  • Short leg = 1, then hypotenuse = 2, long leg = $\sqrt{3}$.
  • If the short leg = $x$, the long leg = $x\sqrt{3}$ and hypotenuse = $2x$.

Practice Notes

• For both triangle types, various practice problems involve using the ratios to find missing side lengths.
• Essential to remember how to derive each leg from the hypotenuse and vice versa using the defined ratios.