Pythagorean Theorem

Definition

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Mathematically, it is expressed as $a^2 + b^2 = c^2$ where $c$ is the hypotenuse.

5 Must Know Facts For Your Next Test

1. In a coordinate plane, the distance between two points can be found using a variation of the Pythagorean Theorem.
2. The theorem can be extended to three dimensions: $a^2 + b^2 + c^2 = d^2$ for a right rectangular prism.
3. The Pythagorean Theorem only applies to right-angled triangles.
4. It provides a fundamental relationship used in trigonometric identities and solving quadratic equations.
5. Many geometric proofs and applications, such as finding areas and perimeters, rely on this theorem.

Review Questions

• What is the formula for the Pythagorean Theorem?
• In which type of triangle does the Pythagorean Theorem apply?
• How can you use the Pythagorean Theorem to find the distance between two points in a coordinate system?

Related terms

Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.

Midpoint Formula: A formula used to find the midpoint between two points in a coordinate plane: $\left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right)$.

Law of Cosines: An extension of the Pythagorean Theorem that applies to any triangle: $c^2 = a^2 + b^2 - 2ab\cos(C)$.