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What Is the Pythagorean Theorem Formula?

The legs and the hypotenuse in a right triangle are connected through the Pythagorean theorem.

Right triangle with legs and hypotenuse


Pythagorean Theorem

The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the legs:

a2 + b2 = c2


The Two Uses of the Pythagorean Theorem

The Pythagorean theorem is mainly used for two purposes:

When you have a right triangle, you are going to find the length of one of the sides of the triangle, and you know the lengths of the other two sides.
When you have an arbitrary triangle, and you want to determine whether it’s a right triangle by testing if a2 + b2 is the same as c2.

So, the theorem is used to find the sides of a right triangle, and to check whether a given triangle is a right triangle.

Example 1

You have a right triangle where AB = 3, BC = 5 and A = 90°. What is the length of AC?

The hypotenuse is always the side of a right triangle opposite to the right angle—which is 90°. You know, therefore, that AC is a leg of the triangle. You put this into the formula and find that AB2 + AC2 = BC2 32 + AC2 = 52 9 + AC2 = 25 AC2 = 25 9 AC2 = 16 AC = 16 AC = 4

The length of side AC is 4.

Example 2

You want figure out if a triangle with sides 5, 8 and 9 is a right triangle.

You know that the hypotenuse is the longest side of a right triangle, so you find the square of the longest side:

92 = 81

Then, you find the sum of the squares of the other sides of the triangle

82 + 52 = 64 + 25 = 89

Since 8189, you know for sure that this isn’t a right triangle.

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