Understanding the Pythagorean Theorem

Aug 3, 2024

Notes on The Pythagorean Theorem

Introduction

  • Welcome to Math Antics!
  • Topic: The Pythagorean Theorem
  • A theorem is a proven statement in math.
  • Pythagoras: Ancient Greek mathematician credited for the theorem.

Understanding the Basics

  • Pre-requisites:
    • Knowledge of angles and triangles
    • Understanding of exponents and square roots
    • Basic algebra, including variables and solving equations
  • Geometry Focus:
    • The theorem applies only to right triangles.
    • Right triangles have one right angle marked with a square symbol.

Components of Right Triangles

  • Hypotenuse:
    • Longest side, opposite the right angle, represented as 'c'.
  • Legs:
    • Two sides forming the right angle, represented as 'a' and 'b'.
  • It doesn't matter which leg is assigned 'a' or 'b' as long as they are consistent.

The Pythagorean Theorem Statement

  • Formula: a² + b² = c²
    • Squaring means multiplying the number by itself.
    • The theorem shows a relationship between the sides of a right triangle.

Example: The 3-4-5 Triangle

  • Triangle sides: 3, 4, 5.
  • Calculating Areas:
    • a = 3: 3² = 9 (Area = 9)
    • b = 4: 4² = 16 (Area = 16)
    • c = 5: 5² = 25 (Area = 25)
  • Check: 9 + 16 = 25 (True)
  • Illustrates geometric and arithmetic relation of the theorem.

Practical Applications

  • Finding Unknown Side Lengths:
    • Use when two sides are known.
    • Example: Triangle with sides 2 cm and 3 cm.
      • 2² + 3² = c²
      • 4 + 9 = c² → c² = 13 → c = √13 cm.
  • Example: Triangle with hypotenuse 6 m and one side 4 m.
    • 4² + b² = 6²
    • 16 + b² = 36 → b² = 20 → b = √20 m or 2√5 m.

Finding Diagonal in a Unit Square

  • Unit square with sides of 1 unit.
  • Diagonal = √(1² + 1²) = √2.

Testing Right Triangles

  • Use the theorem to confirm if a triangle is right.
  • Example: Triangle with sides 3 cm, 3 cm, and 4 cm.
    • Check: 3² + 3² = 9 + 9 = 18 ≠ 4² (16) → Not a right triangle.

Conclusion

  • The Pythagorean Theorem is useful for finding unknown sides and verifying right triangles.
  • Requires understanding of other math skills (geometry, algebra).
  • Practice is essential to mastering these concepts.
  • Visit www.mathantics.com for more resources.