Triangles - Special Right Angles

45°-45°-90° Triangles

• In a 45°-45°-90° triangle:
  • Legs are congruent.
  • If leg = $x$, then hypotenuse = $x\sqrt{2}$.

Find Missing Variables (Examples)

1. Given: leg = 8
   • Find:
     • Hypotenuse = $8\sqrt{2}$.
2. Given: leg = $x$
   • To find:
     • $x = 8$, Hypotenuse = $x\sqrt{2}$.

30°-60°-90° Triangles

• In a 30°-60°-90° triangle:
  • Shorter leg = $x$ (opposite the 30° angle).
  • Longer leg = $x\sqrt{3}$ (opposite the 60° angle).
  • Hypotenuse = $2x$.

Find Missing Variables (Examples)

3. Given: shorter leg = 19
   • Find:
     • Longer leg = $19\sqrt{3}$; Hypotenuse = 38.
4. Given: shorter leg = 5
   • Find:
     • Longer leg = $5\sqrt{3}$; Hypotenuse = 10.
5. Given: longer leg = 25
   • Find:
     • Shorter leg = $\frac{25}{\sqrt{3}}$ (simplified).

Important Relationships

• For 45°-45°-90°:
  • Hypotenuse is $\sqrt{2}$ times the leg.
• For 30°-60°-90°:
  • Hypotenuse is 2 times the shorter leg; longer leg is $\sqrt{3}$ times the shorter leg.