Pythagorean Theorem Lab

For any right triangle with legs a and b and hypotenuse c: a² + b² = c².

To find c, compute the square root of a² + b². For example, with a = 3 and b = 4: 9 + 16 = 25, so c = 5.

You can rearrange to find a missing leg: a = sqrt(c² - b²) or b = sqrt(c² - a²).

A Pythagorean triple is a set of three whole numbers that satisfy a² + b² = c².

Classic triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25.

Any whole-number multiple of a triple is also a triple. The 6-8-10 triple is double 3-4-5. Scaling all three sides by the same factor preserves the relationship.

The area of the square drawn on the hypotenuse equals the combined area of the squares on the two legs.

This geometric interpretation was known to ancient Greek mathematicians including Euclid, and to Babylonian and Chinese scholars even earlier.

The theorem applies only to right triangles. For non-right triangles, the equality becomes an inequality: a² + b² is either less than or greater than c² depending on the angle.

1. For the 3-4-5 triangle: 3² = 9, 4² = 16, 9 + 16 = 25 = 5². Does this match c²?

2. How does 6-8-10 relate to 3-4-5? What happens when you multiply all three sides by 2?

3. If a = 5 and b = 12, what is c? Show your calculation using a² + b² = c².

4. Can you find any triple where a and b are equal (an isosceles right triangle)? What would the ratio a to c be?