Overview
This lecture introduces the Pythagorean theorem, explains its relationship to right triangles, and demonstrates solving for missing side lengths using examples.
Introduction to the Pythagorean Theorem
- The Pythagorean theorem applies only to right triangles.
- It describes the relationship between the sides of a right triangle: the sum of the legs squared equals the hypotenuse squared.
- It is named after Pythagoras, a Greek philosopher and mathematician.
Parts of a Right Triangle
- The hypotenuse is the longest side and is always opposite the right angle.
- The other two sides are called legs; either can be labeled as "a" or "b".
- Conventionally, the hypotenuse is labeled "c".
The Pythagorean Theorem Formula
- The formula is ( a^2 + b^2 = c^2 ), where a and b are legs, and c is the hypotenuse.
- It does not matter which leg is labeled a or b.
Example 1: Finding the Hypotenuse
- Given legs of 4 ft and 3 ft, plug values into the formula: ( 4^2 + 3^2 = c^2 ).
- Calculate: ( 16 + 9 = 25 ), so ( c^2 = 25 ).
- Take the square root: ( c = 5 ) ft.
Visual Representation and Concept
- Squares are constructed on each side of the triangle.
- The area of the squares on the legs (16 and 9 square feet) adds up to the area on the hypotenuse (25 square feet).
- This visualization confirms ( a^2 + b^2 = c^2 ).
Example 2: Finding a Missing Leg
- Given a leg of 15 cm and hypotenuse of 17 cm, set up: ( 15^2 + b^2 = 17^2 ).
- Calculate: ( 225 + b^2 = 289 ), so ( b^2 = 64 ).
- Take the square root: ( b = 8 ) cm.
Key Terms & Definitions
- Right Triangle — a triangle with one 90° angle.
- Hypotenuse — the longest side of a right triangle, opposite the right angle.
- Legs — the two shorter sides of a right triangle.
- Pythagorean Theorem — the relation ( a^2 + b^2 = c^2 ) in right triangles.
Action Items / Next Steps
- Practice solving for missing side lengths in right triangles using the Pythagorean theorem.
- Review the definition and identification of legs and hypotenuse in right triangles.