Converse of the Pythagorean Theorem

Q: Why do builders measure 3-4-5 at a corner?

Mark $3$ feet along one wall and $4$ feet along the other, then measure between the marks. If the diagonal is exactly $5$ feet, the converse guarantees the corner is a right angle. It's the converse of the Pythagorean theorem used as a job-site tool.

The Pythagorean theorem says that in a right triangle, the side lengths satisfy $a^2 + b^2 = c^2$ . The converse runs the logic in the other direction: if a triangle's three sides make $a^2 + b^2 = c^2$ true, the triangle must be a right triangle. If they don't, it is not.

That turns the equation into a test. Given any three side lengths, you can decide — without drawing anything or measuring a single angle — whether those sides form a right triangle.

How to run the test

Step 1: find the longest side. That side takes the $c$ spot, because in a right triangle the hypotenuse is always the longest side. In the figure, $a$ and $b$ are the two legs and $c$ is the longest side.

Step 2: square the two shorter sides and add them — that's $a^2 + b^2$ .

Step 3: square the longest side to get $c^2$ and compare. If the two results match, the triangle is a right triangle. If they don't match, it is not a right triangle. There is no in-between: the squares either balance or they don't.

Two mistakes to avoid

First, the test compares squares, never the plain side lengths. Checking whether the two shorter sides add up to the longest side tells you nothing about right angles — you must square before comparing.

Second, the longest side has to be the one in the $c$ spot. Testing $9$ , $12$ , $16$ as $9^2 + 16^2 = 12^2$ is meaningless. Put $16$ on the right: $9^2 + 12^2 = 225$ while $16^2 = 256$ , so that triangle is not a right triangle.

Pythagorean triples

Certain whole-number sets pass the test and show up constantly: $3, 4, 5$ ; $5, 12, 13$ ; $8, 15, 17$ ; $7, 24, 25$ ; and $20, 21, 29$ . These are called Pythagorean triples.

Any multiple of a triple also works — $6, 8, 10$ is just $3, 4, 5$ doubled, and $12, 16, 20$ is $3, 4, 5$ times four. Recognizing a triple lets you answer without doing the arithmetic.

Worked examples

Example 1: the sides pass the test

A triangle has sides $6$ , $8$ , and $10$ . Is it a right triangle?

The longest side is

10

, so it takes the

c

spot

Square and add the two shorter sides

6^2 + 8^2 = 36 + 64 = 100

Square the longest side

10^2 = 100

Both sides equal

100

, so the equation balances

a^2 + b^2 = c^2

Answer: Yes — the sides satisfy $a^2 + b^2 = c^2$ , so it is a right triangle.

Example 2: the sides fail the test

A triangle has sides $5$ , $7$ , and $9$ . Is it a right triangle?

The longest side is

9

, so it takes the

c

spot

Square and add the two shorter sides

5^2 + 7^2 = 25 + 49 = 74

Square the longest side

9^2 = 81

Compare

74 \neq 81

Answer: No — $74 \neq 81$ , so it is not a right triangle.

Example 3: checking a corner in real life

A carpenter frames a corner. One edge measures $9$ inches, the other edge measures $12$ inches, and the diagonal between their ends measures $15$ inches. Is the corner a right angle?

Treat the two edges as the legs and the diagonal as

c

Square and add the two edges

9^2 + 12^2 = 81 + 144 = 225

Square the diagonal

15^2 = 225

Both equal

225

, so the corner forms a right angle

Answer: Yes — the corner is square.

Try one yourself

Common questions

What's the difference between the theorem and its converse?

The theorem starts with a right triangle and concludes that $a^2 + b^2 = c^2$ . The converse starts with three sides that satisfy $a^2 + b^2 = c^2$ and concludes that the triangle is a right triangle. Same equation, opposite direction of reasoning.

Which side goes in the

c

spot?

Always the longest side. In a right triangle the hypotenuse is the longest side, so the converse test only makes sense when the longest side plays the role of $c$ .

What does it mean if

a^2 + b^2

is bigger or smaller than

c^2

Either way, the triangle is not a right triangle. As a bonus fact: if $a^2 + b^2 > c^2$ the triangle is acute, and if $a^2 + b^2 < c^2$ the triangle is obtuse.

Why do builders measure 3-4-5 at a corner?

Mark $3$ feet along one wall and $4$ feet along the other, then measure between the marks. If the diagonal is exactly $5$ feet, the converse guarantees the corner is a right angle. It's the converse of the Pythagorean theorem used as a job-site tool.

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Converse of the Pythagorean Theorem

How to run the test

Two mistakes to avoid

Pythagorean triples

Worked examples

Example 1: the sides pass the test

Example 2: the sides fail the test

Example 3: checking a corner in real life

Try one yourself

Common questions

Keep learning

The Pythagorean Theorem

Distance in the Coordinate Plane

Angles of Triangles

Angle Relationships & Parallel Lines