Apply the Pythagorean theorem · lesson

Finding the hypotenuse

💡 In a right triangle, a² + b² = c², where c is the longest side across from the right angle.

A right triangle has one square corner. The two sides that make the corner are the legs, a and b. The longest side is the hypotenuse, c.

The rule is a² + b² = c². Square the legs, add, then take the square root.

If the answer isn't a whole number, round it as the question asks.

🧠 Let's work it out

The legs of a right triangle are 20 and 21. Find the hypotenuse.

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💡 To find a leg, subtract: leg² = hypotenuse² − other leg².

If you know the hypotenuse and one leg, subtract instead of adding.

In stories, look for the right angle. A ladder against a wall makes one with the ground. The ladder is the hypotenuse.

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The hypotenuse is 41 and one leg is 9. Find the other leg.

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💡 Three sides make a right triangle only if the two shorter sides squared add up to the longest side squared.

The rule works backward too. Square the two shorter sides and add them. Square the longest side.

If the two results match, it is a right triangle. If not, it isn't.

🧠 Let's work it out

Can 9, 40, and 41 be the sides of a right triangle?

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💡 The distance between two points is the hypotenuse of a right triangle whose legs are the x-change and y-change.

Draw a right triangle with the two points at its ends. One leg goes across and one goes up.

The across leg is the change in x. The up leg is the change in y. Then use a² + b² = c².

🧠 Let's work it out

Find the distance from (0, 1) to (9, 13).

Across: 9 − 0 = 9.
Up: 13 − 1 = 12.
Square and add: 9² + 12² = 81 + 144 = 225.
Take the square root: √225 = 15. The distance is 15.
With negatives, subtract carefully: from x = −4 to x = 3 is 3 − (−4) = 7 across.

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💡 Walking the diagonal of a rectangle is shorter than walking two sides, and the theorem tells you by how much.

A rectangle's diagonal cuts it into two right triangles. The diagonal is the hypotenuse.

Walking two sides means adding them. Compare that with the diagonal to see how much you save.

🧠 Let's work it out

A field is 20 meters by 21 meters. How much shorter is the diagonal than walking two sides?

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