The Pythagorean Theorem describes the exact relationship among the three sides of any right triangle. If the legs have lengths a and b and the hypotenuse has length c, then a² + b² = c². This theorem matters because it connects geometry, measurement, distance, construction, navigation, and coordinate graphs.
A proof shows that the formula is not just a pattern from examples, but a logical result that must always be true for right triangles.
Visual proofs often compare areas because squares built on side lengths have areas a², b², and c². In a rearrangement proof, the same pieces are placed in two different ways, so the leftover areas must be equal. In a similar triangles proof, the altitude to the hypotenuse creates smaller right triangles with matching angles, leading to side ratios that combine into c² = a² + b².
Garfield's trapezoid proof uses the area of a trapezoid made from two right triangles and one isosceles right triangle to derive the same equation.
Key Facts
- For a right triangle with legs a and b and hypotenuse c, a² + b² = c².
- The hypotenuse c is always the side opposite the 90° angle and is the longest side.
- Area of a square on side a is a², on side b is b², and on side c is c².
- Rearrangement proofs work because moving shapes without stretching them preserves total area.
- Similar triangle proof gives a² = c·x and b² = c·y, where x + y = c, so a² + b² = c².
- Garfield's trapezoid proof uses A = 1/2(a + b)(a + b) and also A = 1/2ab + 1/2ab + 1/2c².
Vocabulary
- Right triangle
- A triangle with one angle equal to 90 degrees.
- Leg
- One of the two sides that form the right angle in a right triangle.
- Hypotenuse
- The side opposite the right angle in a right triangle, and the longest side.
- Area proof
- A proof that shows two expressions are equal by showing they represent the same total area.
- Similar triangles
- Triangles with the same angle measures and proportional corresponding side lengths.
Common Mistakes to Avoid
- Using c for a leg instead of the hypotenuse is wrong because c must represent the side opposite the 90° angle.
- Writing a + b = c is wrong because the theorem relates the squares of the side lengths, not the side lengths directly.
- Applying a² + b² = c² to a non-right triangle is wrong because the Pythagorean Theorem only applies when one angle is exactly 90°.
- Forgetting to take the square root when solving for a side is wrong because c² is an area value, while c is a length.
Practice Questions
- 1 A right triangle has legs 6 cm and 8 cm. Use a² + b² = c² to find the hypotenuse.
- 2 A right triangle has hypotenuse 13 m and one leg 5 m. Find the missing leg.
- 3 Explain why a rearrangement proof can show a² + b² = c² without measuring the triangle's sides.