DWG NO. 08.1 — Lesson 1 of 5

The Pythagorean Theorem and Its Converse

Unit 8: Right Triangles and Trigonometry · ~15–30 min

Objective

Find a missing side of a right triangle using the Pythagorean Theorem, and use its converse to classify a triangle as right, acute, or obtuse from its three side lengths.

The Pythagorean Theorem

In any right triangle, the two shorter sides — the legs — meet at the right angle. The longest side, opposite the right angle, is the hypotenuse. The Pythagorean Theorem relates all three: if the legs measure a and b and the hypotenuse measures c, then

a² + b² = c²

This only works for right triangles, and c must always be the hypotenuse — the longest side. Given any two of the three sides, you can solve for the third.

Legs a and b meet at the right angle; the hypotenuse c is opposite it.

The converse: classifying a triangle by its sides

The Converse of the Pythagorean Theorem lets you go the other direction: if a triangle's three sides satisfy a² + b² = c² (with c the longest side), the triangle must be right. Comparing a² + b² to c² also tells you when a triangle is not right.

If a² + b² = c², the triangle is right.
If a² + b² > c², the triangle is acute — the angle opposite c is smaller than 90°.
If a² + b² < c², the triangle is obtuse — the angle opposite c is larger than 90°.

In every case, let c be the longest of the three given sides before comparing.

Worked Example 1 · Finding the hypotenuse

ProblemA right triangle has legs 5 and 12. Find the hypotenuse c.

1.Apply the theorem: 5² + 12² = c².

2.25 + 144 = 169, so c² = 169.

3.Take the square root: c = 13.

c = 13

Worked Example 2 · Finding a leg

ProblemA right triangle has hypotenuse 10 and one leg 6. Find the other leg b.

1.Apply the theorem: 6² + b² = 10².

2.36 + b² = 100, so b² = 64.

3.Take the square root: b = 8.

b = 8

Worked Example 3 · Classifying a triangle

ProblemA triangle has sides 7, 8, and 12. Classify it.

1.The longest side is 12, so let c = 12, a = 7, b = 8.

2.Compare: 7² + 8² = 49 + 64 = 113, and 12² = 144.

3.Since 113 < 144, a² + b² < c².

The triangle is obtuse.

Guided practice

1.A right triangle has legs 9 and 12. Find the hypotenuse.
Show answer
9² + 12² = 81 + 144 = 225, so c = \(\sqrt{225}\) = 15.
2.A right triangle has hypotenuse 17 and one leg 8. Find the other leg.
Show answer
8² + b² = 17², so 64 + b² = 289, b² = 225, b = 15.
3.Classify the triangle with sides 6, 7, and 9.
Show answer
6² + 7² = 36 + 49 = 85, and 9² = 81. Since 85 > 81, the triangle is acute.
4.Classify the triangle with sides 5, 6, and 10.
Show answer
5² + 6² = 25 + 36 = 61, and 10² = 100. Since 61 < 100, the triangle is obtuse.
5.A ladder leans against a wall, reaching 15 ft up the wall with its base 8 ft from the wall. How long is the ladder?
Show answer
The wall and ground form a right angle, so 8² + 15² = c²: 64 + 225 = 289, c = \(\sqrt{289}\) = 17. The ladder is 17 feet long.

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