Math with Mr. J: Introduction to the Pythagorean Theorem

Overview

• The Pythagorean Theorem relates to right triangles and the relationship between their sides.
• Named after Pythagoras, a Greek philosopher and mathematician.
• Applies only to right triangles.
• Fundamental equation: a² + b² = c².

Key Concepts

Sides of a Right Triangle

• Hypotenuse: The longest side, opposite the right angle.
• Legs: The other two shorter sides.

Pythagorean Theorem

• States that the sum of the squares of the legs equals the square of the hypotenuse.
• Equation: a² + b² = c²
  • c: Hypotenuse
  • a and b: Legs (order does not matter)

Example 1: Solving for the Hypotenuse

• Given: Legs are 4 feet and 3 feet.
• Setup: Plug into the equation a² + b² = c²
  • (4)² + (3)² = c²
  • 16 + 9 = 25 = c²
• Solve: Take the square root of both sides.
  • √25 = 5
  • Hypotenuse (c) = 5 feet.

Visual Representation

• Squares on each side of the triangle demonstrate the theorem.
• Areas of the squares on the legs add up to the area of the square on the hypotenuse.
  • (4)² = 16 square feet
  • (3)² = 9 square feet
  • (5)² = 25 square feet
• Verification: 16 + 9 = 25

Example 2: Solving for a Missing Leg

• Given: One leg is 15 cm, hypotenuse is 17 cm.
• Setup: Plug into the equation a² + b² = c²
  • (15)² + b² = (17)²
  • 225 + b² = 289
• Solve: Isolate b²
  • Subtract 225 from both sides: b² = 64
  • Take the square root: √64 = 8
  • Missing leg (b) = 8 cm.

Conclusion

• The Pythagorean Theorem is a reliable method for solving the side lengths of right triangles.
• Helps understand how the areas of squares on the sides relate to each other.
• Equally applicable for finding hypotenuse or a missing leg.