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Use the Pythagorean theorem on right triangles

Find a missing side of a right triangle, estimate a square root without a calculator, find the distance between two grid points, and check whether a corner is square. Includes six original questions with worked answers.

By FinishHigh · Published 2026-11-02 · Updated 2026-11-02

Which triangle, and which side?

The Pythagorean theorem works only on a right triangle: a triangle with one square corner (90 degrees). The side across from the square corner is the hypotenuse. It is always the longest side, and in the formula it is called c. The two sides that make the square corner are the legs, a and b.

The rule is a² + b² = c². It is on the GED® math formula sheet, so you do not need to memorize it. What you do need is to spot which side is c before you put numbers in.

Find the long side: add, then take the square root

A ladder leans against a wall. Its foot is 6 feet from the wall, and it touches the wall 8 feet up. The wall and ground make the square corner, so the ladder is c. 6² + 8² = 36 + 64 = 100, and the square root of 100 is 10. The ladder is 10 feet long.

A TV screen is 24 inches wide and 18 inches tall. Screen size is measured on the diagonal. 24² + 18² = 576 + 324 = 900, and the square root of 900 is 30, so it is a 30-inch screen.

Find a shorter side: subtract

If you know c and one leg, subtract instead of add. A 13-foot ladder stands with its foot 5 feet from a wall. How high does it reach? 13² − 5² = 169 − 25 = 144, and the square root of 144 is 12. It reaches 12 feet up.

Quick check: a leg must come out shorter than c. If your leg is longer than the hypotenuse, you added when you should have subtracted.

When the answer is not a whole number

Legs of 4 and 7 give c² = 16 + 49 = 65. No whole number squared is 65. With the calculator (allowed on Part 2), √65 ≈ 8.06.

Without a calculator, estimate: 8² = 64 and 9² = 81, so √65 is between 8 and 9, very close to 8. That is often enough to pick the right answer choice or to catch a typing mistake.

Distance between two points on a grid

To find the straight-line distance between (1, 2) and (7, 10), draw the right triangle under it. The run is 7 − 1 = 6 and the rise is 10 − 2 = 8. Then 6² + 8² = 100, so the distance is 10 units.

This works for any two points. Subtract the x-values for one leg and the y-values for the other. A negative difference does not matter, because squaring makes it positive.

Is that corner really square?

Builders use the theorem backward. If the three sides fit a² + b² = c², the triangle has a right angle. Sides of 9, 12 and 15: 81 + 144 = 225, and 15² = 225, so yes.

Sides of 7, 8 and 11: 49 + 64 = 113, but 11² = 121. They do not match, so that corner is not square.

Where this comes from

Where this comes from: GED Testing Service’s Assessment Guide for Educators: Mathematical Reasoning lists indicator Q.4.e, using the Pythagorean theorem to find unknown side lengths in a right triangle. The GED® math formula sheet lists a² + b² = c². We checked both on November 2, 2026.

These sources set the topic. The explanations, numbers and questions here are original FinishHigh practice with invented situations, not GED® test questions. GED® is a registered trademark of the American Council on Education. FinishHigh is independent and not affiliated with or endorsed by it. This lesson teaches one skill. It is not a full course or a readiness result.

Try the questions below. For each one, mark the square corner and label c before you calculate.

Try this next: 6 questions ↓

ORIGINAL PRACTICE

Your turn.

Try each question before opening the explanation. These are FinishHigh practice examples, not GED® test questions.

A 10-foot ladder has its foot 6 feet from a wall. How high up the wall does it reach?
Show the answer and why
8 feet. The ladder is c, so subtract: 10² − 6² = 100 − 36 = 64, and √64 = 8.
A park is a rectangle 120 meters long and 50 meters wide. How long is a path straight across, corner to corner? How much shorter is it than walking along two sides?
Show the answer and why
130 meters. 120² + 50² = 14,400 + 2,500 = 16,900, and √16,900 = 130. Walking two sides is 120 + 50 = 170 meters, so the path saves 40 meters.
What is the distance between the points (−2, 1) and (4, 9)?
Show the answer and why
10 units. Run: 4 − (−2) = 6. Rise: 9 − 1 = 8. Then 6² + 8² = 36 + 64 = 100, and √100 = 10.
A right triangle has legs of 5 and 9. Without a calculator, between which two whole numbers is the hypotenuse?
Show the answer and why
Between 10 and 11. c² = 25 + 81 = 106. Since 10² = 100 and 11² = 121, √106 is a little more than 10 (about 10.30).
Is a triangle with sides 8, 15 and 17 a right triangle?
Show the answer and why
Yes. 8² + 15² = 64 + 225 = 289, and 17² = 289. The sides fit a² + b² = c².
A TV has a 40-inch diagonal and is 32 inches wide. How tall is the screen?
Show the answer and why
24 inches. The diagonal is c: 40² − 32² = 1,600 − 1,024 = 576, and √576 = 24.