Special Right Triangles

Overview

• Special right triangles are frequently used due to their unique properties.
• Two main types: 45-45-90 and 30-60-90 triangles.

45-45-90 Right Triangle

• Formation: Derived from cutting a square in half diagonally.
• Angles: 45°, 45°, and 90°.
• Properties:
  • Isosceles triangle: Both legs have equal length.
  • If one leg is 'a', the other leg is also 'a'.
  • Hypotenuse is a√2.

Calculating Hypotenuse

• Use Pythagorean Theorem: 1.
  • Equation: a² + a² = c²
  • Simplified: 2a² = c²
  • Hypotenuse (c): c = a√2
• Shortcut: Multiply the length of a leg by √2 to get the hypotenuse.

Examples

• If leg = 5, hypotenuse = 5√2
• If leg = 7, hypotenuse = 7√2

30-60-90 Right Triangle

• Formation: Derived by cutting an equilateral triangle in half.
• Angles: 30°, 60°, and 90°.

Properties

• Shortest leg is half the hypotenuse.
• Relationship between sides:
  • Hypotenuse = 2 × (shorter leg)
  • Longer leg = √3 × (shorter leg)

Examples

• If shortest leg = 4, hypotenuse = 8, longer leg = 4√3
• If hypotenuse = 12, shortest leg = 6, longer leg = 6√3

Identifying Triangles

• Determine if a given triangle is a 45-45-90, 30-60-90, or neither by analyzing side lengths:
  • Presence of √2 implies a 45-45-90 triangle.
  • Presence of √3 with proper doubling relation implies a 30-60-90 triangle.
  • Neither pattern: Not a special triangle.

Examples

1. Side lengths with doubling and √3 suggest a 30-60-90 triangle.
2. Side lengths with identical values and √2 suggest a 45-45-90 triangle.
3. Absence of these patterns suggests neither type.

Practice Problems

• Engage with practice problems to solidify understanding of identifying and calculating aspects of special right triangles.