Detailed definition
Understanding Pythagorean Theorem
Pythagorean Theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. In a right triangle, a squared plus b squared equals c squared. In the familiar notation, if the legs are a and b and the hypotenuse is c, then a squared plus b squared equals c squared.
The theorem works only for right triangles. That condition is not a small detail; it is the whole reason the side relationship holds. Without the ninety-degree angle, the formula no longer describes the triangle correctly.
This theorem is used constantly in distance, coordinate geometry, construction, and trigonometry preparation. The converse is also useful: if three side lengths satisfy the equation, the triangle is right.
Key facts
Important ideas to remember
- In a right triangle, a squared plus b squared equals c squared.
- The theorem applies only to right triangles.
- The hypotenuse is the side opposite the right angle and is the side whose square stands alone in the equation.
- The converse allows side lengths to prove that a triangle is right.
Where it is used
Where pythagorean theorem shows up
- Use the Pythagorean Theorem to find a missing side in a right triangle.
- Use it in coordinate geometry to derive distance between points.
- Use it in real-world measurement problems involving perpendicular directions.
Common mistakes
What to watch out for
- Do not apply the theorem to a triangle unless a right angle is present or proven.
- Do not put the wrong side in the hypotenuse position of the equation.
- Do not forget to square the side lengths before adding or comparing them.