The Pythagorean theorem says right triangles satisfy $a^{2} + b^{2} = c^{2}$. The converse runs the other way: a triangle whose sides satisfy it must have a right angle. Builders use this to check square corners.
Why it works
Suppose a triangle has sides $a$, $b$, $c$ with $a^{2} + b^{2} = c^{2}$. Build a right triangle with legs $a$ and $b$. By the theorem, its hypotenuse is $\sqrt{a^{2} + b^{2}} = c$. So the two triangles have the same three side lengths. Three sticks can be joined into a triangle in only one shape, so the triangles are congruent, and the first triangle has the same right angle.
Key idea
Converse: if the sides of a triangle satisfy $a^{2} + b^{2} = c^{2}$, where $c$ is the longest side, then the triangle has a right angle, opposite $c$. If they don’t, it has no right angle.
Worked examples
Worked example: A right triangle
Is a triangle with sides 9, 40 and 41 a right triangle?
- The longest side is 41: $41^{2} = 1681$.
- $9^{2} + 40^{2} = 81 + 1600 = 1681$.
Answer: Yes
Worked example: Not a right triangle
Is a triangle with sides 6, 8 and 11 a right triangle?
- $6^{2} + 8^{2} = 100$, but $11^{2} = 121$.
- They are not equal.
Answer: No
Worked example: A square corner
A builder measures 3 ft along one wall and 4 ft along the other, and the diagonal between those marks is 5 ft. Is the corner square?
- $3^{2} + 4^{2} = 25 = 5^{2}$.
- By the converse, the corner is a right angle.
Answer: Yes
If $c^{2}$ is greater than $a^{2} + b^{2}$, the angle across from $c$ is wider than $90^\circ$ (as with 6, 8, 11). If it is less, the angle is narrower.
Common mistake
Testing with a side that isn’t the longest as $c$. For 15, 8, 17, the test is $8^{2} + 15^{2} = 17^{2}$ (true), not $8^{2} + 17^{2} = 15^{2}$. Always put the longest side alone on one side.