Lesson 2: Understand the Converse of the Pythagorean Theorem

Property

For a triangle with side lengths $a, b,$ and $c$, if the sides satisfy the equation $a^2 + b^2 = c^2$, then the triangle is a right triangle. The right angle is always opposite the longest side, $c$.

Examples

• A triangular garden has sides measuring 8 meters, 15 meters, and 17 meters. Is it a right triangle? Check: $8^2 + 15^2 = 64 + 225 = 289$. The longest side squared is $17^2 = 289$. Yes, it's a right triangle.

• A carpenter builds a frame with sides 5 ft, 12 ft, and a diagonal of 13 ft. Since $5^2 + 12^2 = 25 + 144 = 169$, and $13^2 = 169$, the frame must have a 90-degree corner.

Property

To prove the Converse of the Pythagorean Theorem, we use an indirect proof involving a second triangle.

Given a triangle $\Delta ABC$ with side lengths $a$, $b$, and $c$ where $a^2 + b^2 = c^2$, we construct a separate right triangle, $\Delta DEF$, with legs of length $a$ and $b$.

Property

To determine if side lengths $a$, $b$, and $c$ (where $c$ is the longest side) form a right triangle, you must verify two conditions in order:

1. Triangle Inequality Theorem: The sides can form a triangle.
   $$a + b > c$$
2. Converse of the Pythagorean Theorem: The triangle is a right triangle.
   $$a^2 + b^2 = c^2$$

Examples