Lesson 7-4: 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

A Pythagorean triple is a set of three positive integers $a$, $b$, and $c$ that perfectly satisfy the Pythagorean Theorem:

Common base triples include:

• $3, 4, 5$
• $5, 12, 13$
• $8, 15, 17$
• $7, 24, 25$

Property

In real-world applications like construction, measurements are often approximate. To determine if a physical corner is a right angle (often called being "square"), check if the sum of the squares of the two shorter sides is approximately equal to the square of the diagonal:

If the values are exactly equal or fall within an acceptable measurement tolerance, the corner is considered a right angle.

Examples