Right Triangles & Trigonometry

The Pythagorean Theorem

Applying the Pythagorean Theorem and its converse to right triangles.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Working the Theorem in Reverse

You already know that for a right triangle, a² + b² = c². Before reading on, consider the reverse question: if you're handed three side lengths and confirm that the two smaller ones squared and added really do equal the largest one squared, does that guarantee the triangle is a right triangle — or could a non-right triangle also happen to satisfy that equation?

Definition — The Converse of the Pythagorean Theorem

For a triangle with sides a, b, and c (c the longest): if a² + b² = c², the triangle is a right triangle. This converse is also true, letting you classify a triangle as right, acute, or obtuse purely from its side lengths, with no angle measurement needed.

Comparing c² to a² + b² actually classifies any triangle, not just right ones: if c² equals a² + b², the triangle is right; if c² is less than a² + b², the angle opposite the longest side is smaller than 90°, making the triangle acute; if c² is greater than a² + b², that angle is larger than 90°, making the triangle obtuse.

Beyond the area-rearrangement proof from earlier grades, similarity offers another way to see why the theorem holds. Drawing the altitude from a right triangle's right angle down to the hypotenuse splits it into two smaller triangles — and both of those smaller triangles turn out similar to the original triangle and to each other, since all three share the same set of angles (each contains the original right angle's complement in some form, by angle-sum reasoning). The proportional side relationships that similarity guarantees between these three nested triangles rearrange algebraically into exactly a² + b² = c².

Worked Example — Applying the Converse to Classify a Triangle

A triangle has sides 7, 8, and 12. Classify it. The longest side is 12, so compare 12² to 7² + 8²: 144 versus 49 + 64 = 113. Since 144 > 113, this triangle is obtuse.

Worked Example — Confirming a Right Triangle

A triangle has sides 9, 12, and 15. Is it a right triangle? Compare 15² to 9² + 12²: 225 versus 81 + 144 = 225. Since they're equal, this is a right triangle by the converse of the Pythagorean Theorem.

a^{2} + b^{2} = c^{2} \;\Longleftrightarrow\; \text{right triangle}

Tip

Always square the longest side and compare it against the sum of the other two squared — comparing sides in the wrong position gives a meaningless result, since the converse's classification specifically depends on which side is the longest.

Common Mistakes

Applying the converse without first identifying which side is the longest, accidentally squaring the wrong side by itself on one side of the comparison.
Always identify the longest side first and isolate it — the classification test specifically compares that side's square to the sum of the other two squares, not any other combination.
Assuming the converse only confirms right triangles, without realizing the same comparison also classifies acute and obtuse triangles.
The comparison between c² and a² + b² gives three possible outcomes, not just one — equal means right, less means acute, and greater means obtuse.

Key Takeaways

The converse of the Pythagorean Theorem confirms a right triangle from side lengths alone, with no angle measurement.
Comparing c² to a² + b² also classifies any triangle as acute, right, or obtuse.
An altitude drawn to a right triangle's hypotenuse creates two smaller similar triangles, offering a second proof of the theorem through proportional reasoning.

Summary

The converse extends the Pythagorean Theorem from measuring known right triangles to classifying unknown ones. The next lesson uses right triangles' proportional structure to define three ratios that work for any acute angle.

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