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Algebra I

Pythagorean Theorem

Lesson

In a right triangle (one with a 90° angle), the two shorter sides are called the legs, and the longest side — directly across from the right angle — is called the hypotenuse.

The Pythagorean theorem says that for any right triangle, if the legs are aa and bb and the hypotenuse is cc:

a2+b2=c2a^2 + b^2 = c^2

This means: square each leg, add them up, and you get the square of the hypotenuse. To find a missing side, plug in what you know and solve.

Worked example 1 — finding the hypotenuse

Legs are 3 and 4. Find the hypotenuse cc.

32+42=c23^2 + 4^2 = c^2
9+16=c29 + 16 = c^2
25=c225 = c^2

Take the positive square root:

c=5c = 5

Worked example 2 — finding a leg

One leg is 6 and the hypotenuse is 10. Find the other leg bb.

62+b2=1026^2 + b^2 = 10^2
36+b2=10036 + b^2 = 100

Subtract 36 from both sides:

b2=64b^2 = 64
b=8b = 8

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

legs 3 and 4, find c\text{legs } 3 \text{ and } 4, \text{ find } c

Problem 2

legs 6 and 8, find c\text{legs } 6 \text{ and } 8, \text{ find } c

Problem 3

legs 5 and 12, find c\text{legs } 5 \text{ and } 12, \text{ find } c

Practice

Standard problems matching the lesson.

Problem 4

legs 8 and 15, find c\text{legs } 8 \text{ and } 15, \text{ find } c

Problem 5

legs 7 and 24, find c\text{legs } 7 \text{ and } 24, \text{ find } c

Problem 6

legs 9 and 12, find c\text{legs } 9 \text{ and } 12, \text{ find } c

Problem 7

leg 6, hypotenuse 10, find other leg\text{leg } 6, \text{ hypotenuse } 10, \text{ find other leg}

Problem 8

leg 9, hypotenuse 15, find other leg\text{leg } 9, \text{ hypotenuse } 15, \text{ find other leg}

Problem 9

leg 5, hypotenuse 13, find other leg\text{leg } 5, \text{ hypotenuse } 13, \text{ find other leg}

Problem 10

leg 8, hypotenuse 17, find other leg\text{leg } 8, \text{ hypotenuse } 17, \text{ find other leg}

Problem 11

legs 20 and 21, find c\text{legs } 20 \text{ and } 21, \text{ find } c

Problem 12

leg 11, hypotenuse 61, find other leg\text{leg } 11, \text{ hypotenuse } 61, \text{ find other leg}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 13

legs 9 and 40, find c\text{legs } 9 \text{ and } 40, \text{ find } c

Problem 14

legs 12 and 35, find c\text{legs } 12 \text{ and } 35, \text{ find } c

Problem 15

leg 28, hypotenuse 53, find other leg\text{leg } 28, \text{ hypotenuse } 53, \text{ find other leg}

Problem 16

leg 33, hypotenuse 65, find other leg\text{leg } 33, \text{ hypotenuse } 65, \text{ find other leg}

Problem 17

legs 14 and 48, find c\text{legs } 14 \text{ and } 48, \text{ find } c

Problem 18

legs 15 and 36, find c\text{legs } 15 \text{ and } 36, \text{ find } c