How to Use the Pythagorean Theorem to Find a Leg

Problem

c = 13,\quad b = 12

Answer

a = 5

The Pythagorean theorem usually finds the hypotenuse, but it works backwards too. Subtract the known leg squared from the hypotenuse squared, then take the root.

Step-by-step solution

Write the Pythagorean theorem.
$a^2 + b^2 = c^2$
Why: The theorem links the two legs and the hypotenuse of a right triangle.
Substitute the known values.
$a^2 + 12^2 = 13^2$
Why: The hypotenuse, 13, takes the place of c.
Subtract the known square.
$a^2 = 169 - 144 = 25$
Why: Rearrange the theorem to isolate the unknown square.
Take the positive square root.
$a = \sqrt{25} = 5$
Why: A length cannot be negative, so only the positive root fits.

Common mistakes

Adding 169 + 144 instead of subtracting. Finding a leg means subtracting.
Putting 12 on the c side. The hypotenuse is always the longest side.
Reporting -5. Lengths are positive.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

$c = 10,\quad b = 6$

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$a = 8$
$c = 15,\quad b = 9$

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$a = 12$
$c = 25,\quad b = 7$

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$a = 24$
$c = 17,\quad b = 8$

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$a = 15$
$c = 26,\quad b = 10$

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$a = 24$