Calculating Missing Side Lengths of a Triangle

Introduction

• Discusses methods to calculate missing side lengths in triangles.
• Covers methods from basic to advanced.

Pythagorean Theorem

• Formula: a² + b² = c² where c is the hypotenuse.

• If hypotenuse (c) is known, it is the longest side, opposite the 90-degree angle.

• Example 1: Right triangle with sides 6, x, and hypotenuse 10.

  • Solve for x:
    • 6² + x² = 10²
    • 36 + x² = 100
    • x² = 100 - 36
    • x = √64 = 8

• Example 2: Right triangle with hypotenuse 16 and one leg 9.

  • Solve for x:
    • 9² + x² = 16²
    • 81 + x² = 256
    • x² = 256 - 81
    • x² = 175 (simplified to 5√7)

Trigonometric Ratios

• SOHCAHTOA: Sine, Cosine, Tangent
• Used when angles and one side are known.

Example Using Sine

• Right triangle with angle 30°, hypotenuse 50, find opposite side.
  • Sine 30 = opposite/hypotenuse
  • Sin 30 = x/50
  • Solve for x: x = 50 * 0.5 = 25*

Example Using Tangent

• Right triangle with angle 60°, one side 100, find other side.
  • Tangent 60 = 100/x
  • Solve for x using Tangent value: x = 100/√3, rationalized to 100√3/3

Law of Sines

• Used for non-right triangles with known angles and sides.
• Formula: Sin A/a = Sin B/b = Sin C/c

Example

• Triangle with angles A = 50, B = 40, side b = 10, find side a.
  • Sin 50/x = Sin 40/10
  • Solve for x: x ≈ 11.9

Law of Cosines

• Used for non-right triangles with two sides and included angle.
• Formula: c² = a² + b² - 2ab * Cos(angle)*

Example

• Triangle with sides 9, 10, angle 30°, find third side.
  • Apply formula and solve: x ≈ 5.01

Composite Triangles

• Approach using combinations of learned methods.
• Example involves finding missing sides in a composite triangle.
  • Set up equations using Pythagorean theorem and trigonometric ratios.
  • Solve using algebraic methods.

Geometric Mean

• Used in specific configurations of right triangles (e.g., altitude to hypotenuse).
• Example: Geometric mean of segments 8 and 32.
  • Calculation: x = √(832) = 16

Conclusion

• Multiple methods are available to calculate missing sides:
  • Pythagorean theorem for right triangles.
  • SOHCAHTOA for trigonometric ratios.
  • Law of Sines and Cosines for non-right triangles.
  • Geometric Mean for specific configurations.
• Algebraic manipulation and understanding of properties are key.