7.2 Pythagorean theorem & applications

The Pythagorean Theorem

A right triangle has one 90-degree angle. The two sides that form the right angle are the legs (labeled a and b), and the side opposite the right angle is the hypotenuse (labeled c). The hypotenuse is always the longest side. The Pythagorean theorem connects the three sides:

a^2 + b^2 = c^2

In words: the sum of the squares of the two legs equals the square of the hypotenuse. This relationship works only for right triangles, and it is one of the most heavily tested ideas on the PERT.

Finding the hypotenuse

When you know both legs, square them, add the results, and take the square root.

Worked example. A right triangle has legs 6 and 8. Find the hypotenuse.

a^2 + b^2 = c^2 6^2 + 8^2 = c^2 36 + 64 = c^2 100 = c^2 c = sqrt(100) = 10.

The hypotenuse is 10. Notice you take the positive square root because a length can never be negative.

Finding a missing leg

If you know the hypotenuse and one leg, rearrange the formula and subtract.

Worked example. A right triangle has hypotenuse 13 and one leg 5. Find the other leg.

a^2 + b^2 = c^2 5^2 + b^2 = 13^2 25 + b^2 = 169 b^2 = 169 - 25 = 144 b = sqrt(144) = 12.

The missing leg is 12. The key move is subtracting the known leg's square from the hypotenuse's square. A common trap is to add when you should subtract, so always check which side is the hypotenuse first.

Common Pythagorean triples

A Pythagorean triple is a set of three whole numbers that fit a^2 + b^2 = c^2. Memorizing a few lets you answer some questions instantly, without a calculator. Any whole-number multiple of a triple is also a triple.

Worked example. A right triangle has legs 9 and 12. Because 9-12-15 is 3 times the 3-4-5 triple, the hypotenuse is 15. Check: 9^2 + 12^2 = 81 + 144 = 225 = 15^2. Correct.

Checking whether a triangle is a right triangle

The theorem also works in reverse (the converse): if the three side lengths satisfy a^2 + b^2 = c^2, the triangle must be a right triangle. Always let c be the longest side.

Worked example. Do sides 8, 15, and 17 form a right triangle? The longest side is 17, so test whether 8^2 + 15^2 equals 17^2. Left side: 64 + 225 = 289. Right side: 17^2 = 289. The two sides match, so yes, it is a right triangle (and 8-15-17 is a triple).

Worked example. Do sides 4, 5, and 6 form a right triangle? Test 4^2 + 5^2 against 6^2: 16 + 25 = 41, but 6^2 = 36. Since 41 does not equal 36, this is not a right triangle.

Real-world applications

Word problems often hide a right triangle inside a picture. Draw the triangle, label the legs and hypotenuse, then apply the theorem.

The ladder problem. A 10-foot ladder leans against a wall. Its base is 6 feet from the wall. How high up the wall does the ladder reach? The ladder is the hypotenuse (c = 10), the distance from the wall is one leg (a = 6), and the height up the wall is the other leg (b).

6^2 + b^2 = 10^2 36 + b^2 = 100 b^2 = 100 - 36 = 64 b = sqrt(64) = 8.

The ladder reaches 8 feet up the wall. This is just the 6-8-10 triple in action.

The diagonal of a rectangle. A television screen is 12 inches tall and 16 inches wide. How long is the diagonal? The height and width are the legs; the diagonal is the hypotenuse.

12^2 + 16^2 = c^2 144 + 256 = c^2 400 = c^2 c = sqrt(400) = 20.

The diagonal is 20 inches, which is exactly how screens are advertised, using that diagonal measurement.

Distance across a park. Instead of walking 300 feet east and then 400 feet north, you cut straight across. How far is the shortcut?

300^2 + 400^2 = c^2 90,000 + 160,000 = c^2 250,000 = c^2 c = sqrt(250,000) = 500.

The straight path is 500 feet, a 3-4-5 triple scaled by 100. Walking around the two sides took 700 feet, so the diagonal shortcut saves 200 feet.

A quick strategy

For any right-triangle problem: (1) identify the hypotenuse, which is opposite the right angle and is the longest side; (2) decide whether you are finding the hypotenuse (add the squares, then take the square root) or a leg (subtract, then take the square root); (3) check whether the numbers form a known triple to save time. Careful setup, especially knowing which side is c, prevents almost every error on these questions. Finally, sanity-check your answer: the hypotenuse must be longer than either leg, so if you solve for c and get a number smaller than a leg, you added or subtracted in the wrong direction and should redo the step.