30-60-90 Right Triangle

Overview

• Important relationships of sides in a 30-60-90 triangle:
  • Side across 30° angle = 1
  • Hypotenuse = 2
  • Side across 60° angle = sqrt(3) times the side across 30° angle
• Key rules:
  • From 30° to 90°: Multiply by 2
  • From 90° to 30°: Divide by 2
  • From 30° to 60°: Multiply by sqrt(3)
  • From 60° to 30°: Divide by sqrt(3)
• Shortest side is always across the smallest angle (30°), and longest side (hypotenuse) is across the largest angle (90°).

Example Calculations:

Example 1

• Given: 30° side = 5
• Calculate other sides:
  • Hypotenuse: 5 * 2 = 10
  • 60° side: 5 * sqrt(3) = 5 sqrt(3)

Example 2

• Given: 30° side = 12 sqrt(3)
• Calculate other sides:
  • Hypotenuse: 12 sqrt(3) * 2 = 24 sqrt(3)
  • 60° side: 12 sqrt(3) * sqrt(3) = 12 * 3 = 36

Example 3

• Given: Hypotenuse = 12
• Calculate other sides:
  • 30° side: 12 / 2 = 6
  • 60° side: 6 * sqrt(3) = 6 sqrt(3)

Example 4

• Given: Hypotenuse = 16 sqrt(3)
• Calculate other sides:
  • 30° side: 16 sqrt(3) / 2 = 8 sqrt(3)
  • 60° side: 8 sqrt(3) * sqrt(3) = 8 * 3 = 24

Example 5

• Given: 60° side = 9
• Calculate other sides:
  • 30° side: 9 / sqrt(3) = 3 sqrt(3)
  • Hypotenuse: 3 sqrt(3) * 2 = 6 sqrt(3)

Example 6

• Given: 60° side = 14 sqrt(3)
• Calculate other sides:
  • 30° side: 14 sqrt(3) / sqrt(3) = 14
  • Hypotenuse: 14 * 2 = 28

SAT Math Question

• Given: AC (hypotenuse) = 20 units
• Area of shaded region = Area of Circle - Area of Triangle
• Circle: Radius = AC / 2 = 20 / 2 = 10
  • Area = π * (10)^2 = 100π
• Triangle:
  • 30° side = 20 / 2 = 10
  • 60° side = 10 * sqrt(3) = 10 sqrt(3)
  • Area = 1/2 * base * height = 1/2 * 10 sqrt(3) * 10 = 50 sqrt(3)
• Area of shaded region = 100π - 50 sqrt(3)
  • Decimal: 227.6 square units