The Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the total of the squares of each of the triangle legs lengths is identical to the square of the triangle’s hypotenuse length.

A long, long time ago, Pythagoras, a Greek mathematician, discovered a fascinating property of right triangles.

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1. Determine the missing length in this right triangle using the Pythagorean theorem. Use this formula: \(a^{2} + b^{2} = c^{2} \)

\(b=12\)
\(c=15\)
\(a=?\)
A.
B.
C.

Question 1 of 2

2. Determine the missing length in this right triangle using the Pythagorean theorem. Use this formula: \(a^{2} + b^{2} = c^{2} \)

\(a=12\)
\(b=16\)
\(c=?\)
A.
B.
C.

Question 2 of 2


 

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Video Transcription

If you add the squares of the triangle’s legs lengths, the outcome is equal to the square of the triangle’s hypotenuse length.

This property that has so many applications in architecture, engineering, art, and science is what we call the “Pythagorean Theorem.”

So let’s see how this theorem is helping you to learn more about how triangles are constructed. And the good thing is that you even don’t need to speak the Greek language to apply or understand the discovery of Pythagoras.

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The Pythagorean Theorem

The Greek mathematician Pythagoras was studying right triangles and in what way the hypotenuse and the legs of right triangles were related before he derived his theory.

 

The Pythagorean Theorem
In case a and b symbolize the lengths of a right triangle’s legs, and c symbolizes the hypotenuse’s length, then the sum of the squares of the leg’s lengths is the same as the square of the hypotenuse’s length.

This interesting relationship is represented in this formula: a² + b² = c²

In this box above, you could have noticed the expression “square,” but also the small “2” s to the top of the letters like in a² + b² = c². To square a number, we multiply that number by itself. If we square, for example, the number “5”, we multiply five times 5; If we square “12”, we multiply 12 times 12. A few common squared numbers are shown in the following table.

When we see the following equation: a² + b² = c², we can see this as follows: “the length of the side with “a” times itself (so a²), plus the length of the side with “b” times itself (so b²) equals the length of the side with “c” times itself so c²).”

Let’s take a look at the Pythagorean Theorem using an example of a right triangle.

The Pythagorean theorem holds true for the right triangle above. The sum of both legs’ length squares (a² plus b²) equals the square of the hypotenuse’s length (c²). This actually is holding true for every right triangle!

This Pythagorean Theorem may be represented in terms of “area” as well.

In all right triangles, areas of squares drawn from any hypotenuse equals the sum of areas of squares drawn from the triangles’ two legs. Just take a look at the following illustration in that same 3-4-5 right triangle.

Please note that this Pythagorean Theorem is ONLY working if we have right triangles.

So again, the Pythagorean Theorem states that in right triangles, the total of the squares of each of the triangle legs lengths is identical to the square of the triangle’s hypotenuse length.

 

Last Updated on June 13, 2022.