Overview
This lecture introduces the Pythagorean Theorem, explaining its application to right triangles and demonstrating how to use it to solve for unknown side lengths.
The Pythagorean Theorem: Basics
- The Pythagorean Theorem applies only to right triangles (triangles with a 90-degree angle).
- The side opposite the right angle is called the hypotenuse and is always the longest side.
- The other two sides are called the legs of the triangle.
- The theorem states: the sum of the squares of the legs equals the square of the hypotenuse.
- The formula is: a² + b² = c², where a and b are the legs, and c is the hypotenuse.
- It does not matter which leg is labeled a or b.
Example 1: Solving for the Hypotenuse
- Given a right triangle with legs of 4 feet and 3 feet.
- Substitute into the formula: 4² + 3² = c²; so 16 + 9 = 25; c² = 25.
- Take the square root: c = 5 feet.
- A visual shows that squares built on the legs (areas 16 and 9) sum to the area built on the hypotenuse (25).
Example 2: Solving for a Missing Leg
- Given a right triangle with one leg of 15 cm and hypotenuse of 17 cm.
- Substitute into the formula: 15² + b² = 17²; so 225 + b² = 289.
- Subtract 225: b² = 64.
- Take the square root: b = 8 cm.
Key Terms & Definitions
- Right Triangle — a triangle with one 90-degree angle.
- Hypotenuse — the longest side of a right triangle, opposite the right angle.
- Legs — the two shorter sides of a right triangle, adjacent to the right angle.
- Pythagorean Theorem — a² + b² = c²; relates the sides of a right triangle.
Action Items / Next Steps
- Practice using the Pythagorean Theorem to solve for missing side lengths in right triangles.
- Review the definitions of hypotenuse and legs.
- Visualize the theorem by drawing squares on each side of a right triangle.