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What Is the Pythagorean Theorem and When Is It Used?

What is the Pythagorean theorem? It’s a trigonometry equation used to find the length of one side of a right triangle. Though similar concepts had been discovered by the Babylonians, Greek Mathematician Pythagoras was the first person to come up with a geometric proof about how the sum of the squares of the lengths can determine the side lengths of a right triangle.

Pythagoras determined that when three squares are arranged so that they form a right angle triangle, the largest of the three squares must have the same area as the other two squares combined. In the picture below, you can see how the sum of the squares creates the right triangle ABC.

What is the Pythagorean theorem: Graph of how the sum of the squares creates the right triangle ABC

This realization about the area of the squares led to the Pythagoras theorem:

What is the Pythagorean theorem: a2+b2=c2 formula

Squares are different from other parallelograms and trapezoids because all their sides are equal lengths. So since squares are made up of four equal sides, you can see that each individual square makes up a side of the right triangle.

The length of the largest square, which we'll call length c, is the length of the hypotenuse. (The hypotenuse is the longest side of a right triangle.) The smaller squares make up the other two sides of the right triangle.

Breaking Down the Pythagorean Theorem

Let's get into the dissection and proof of the Pythagorean theorem. The Pythagorean theorem states:

  • The legs of a right triangle that are adjacent to a 90-degree angle must be squared and added together.

What is the Pythagorean theorem: Formula and proof of the Pythagorean theorem

  • The length of the third side, the hypotenuse of a right triangle, must also be squared.

Formula and proof of the Pythagorean theorem

  • You must then take the square of the length of the hypotenuse and the square root of the other two sides of a right triangle to solve the equation.

Formula and proof of the Pythagorean theorem

The Answer To ‘What Is the Pythagorean Theorem’

The Pythagorean Theorem is a useful tool that shows how the sum of the areas of three intersecting squares can determine the side lengths of a right triangle. This theorem is an extremely useful tool that provides the basis for more complex trigonometry theories such as the converse of the Pythagorean theorem.

This theorem, however, isn't a generalization about all triangles. The Pythagoras theorem doesn't apply to obtuse or isosceles triangles that don't contain right angles.

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