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6th-8th Grade Math: Practice & Review55 chapters | 469 lessons

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Lesson Transcript

Instructor:
*Kevin Newton*

Kevin has edited encyclopedias, taught middle and high school history, and has a master's degree in Islamic law.

One of the most useful aspects of studying triangles is the Pythagorean Theorem. In this lesson, we use it to find the unknown length of a side of a triangle, whether it is a shorter side or the hypotenuse.

Right triangles have long been some of the most interesting shapes that geometry has studied. In fact, as you progress further in math, you'll find that there is an entire subject dedicated to studying the ratios of their angles and lengths. But we're not there yet.

Instead, we're only interested in one ratio that makes use of all three sides of a triangle - the **Pythagorean Theorem**. Named for the ancient Greek philosopher Pythagoras, the theory is pretty simple and straightforward. *a*^2 + *b*^2 will equal *c*^2. Put another way, if you square the two shorter sides of a right triangle and add them together, you'll end up with the square of the hypotenuse.

In this lesson, we'll not only learn more about how it works, but we'll see how we can use it to find both the hypotenuse of a right triangle as well as the measure of one of the shorter sides.

But first, why do we know that the Pythagorean Theorem even works? For that, we're going to need some cubes. 50, to be precise. Make three bigger pieces, one with 9 cubes as a square, another with 15 pieces as a square, and the last with 25 pieces as a square. Got them? Good. Now make a triangle out of them. You'll see that the one with the 25 pieces makes the hypotenuse of the triangle, while the other two make up the sides.

And what is the measure of these sides? 3, 4, and 5. Okay, so it's a useful trick, but how do we know for sure it works? For that, we'll need more cubes. If you want to keep playing with them, try these triangles for example: 6, 8, 10; 7, 14, 15; 33, 44, 55; or, if you're feeling really lucky: 37, 684, 685. Do you really want to count out all those blocks? I didn't think so, so just take my word on it!

So now that we know it works, how do we use it? Simply put, just plug in the numbers that we know. For example, let's prove that massive triangle from earlier - the 37, 684, 685 triangle. Plug them into your formula of *a*^2 + *b*^2 = *c*^2. That means 37^2 + 684^2 = 685^2. In case you were curious, those numbers come out to being 1,369 + 467,856 = 469,225. This happens to add up to 469,225, which means that the triangle in question is definitely a right triangle.

So now that we've used it to prove that a triangle is a right triangle, let's work through a couple of examples that show us how to find either the hypotenuse or one of the sides.

Let's say you were trying to find the hypotenuse of a triangle with sides of 16 and 63. You know it's a right triangle, so how do you find the length of the hypotenuse? Let's plug in what we know. Remember *a* and *b* represent the shorter sides of the triangle, so that means *a*^2 + *b*^2 becomes 16^2 + 63^2 = *c*^2. That reduces to 256 + 3,969 = *c*^2, or 4,225 = *c*^2.

To solve for *c*, we simply take the square root of both sides. The square root of *c*^2 is just *c*, while the square root of 4,225 is 65. That means that the hypotenuse of this triangle has a length of 65.

That was pretty straight-forward, but what about situations where you have the length of the hypotenuse and one of the sides. How do you find the other? In that case, we simply solve for either *a* or *b*. It doesn't matter which one, as you can change your mind about if a side is *a* or *b*, as long as you're consistent.

So let's look at a triangle with a hypotenuse of 41 and a shorter side of 9. It is a right triangle, so how do we find the missing side? Let's go back to our formula: *a*^2 + *b*^2 = *c*^2. That's setup to solve for *c*, but we know *c*. Instead, let's set it up to solve for one of the sides, in this case *a*. Again, it doesn't matter if you choose *a* or *b*.

So, *a*^2 = *c*^2 - *b*^2, since we had to subtract *b*^2 from both sides to get *a*^2 alone. Now plug in your numbers. *c* becomes 41 and *b* becomes 9. In the formula, that means that *a*^2 = 41^2 - 9^2. That simplifies to 1,681 - 81, or *a*^2 = 1,600. Take the square root and you get *a* = 40.

Note that you may not always get such cleanly round numbers as these. That doesn't matter; as long as you follow the procedure, you'll still get the right answer!

In this lesson, we learned about the **Pythagorean Theorem**, as well as how to use it to find either the hypotenuse or the shorter side of a right triangle. Remember that the Pythagorean Theorem states that the sum of the squares of the two short sides of a right triangle is equal to the square of the hypotenuse. Often, we write this as *a*^2 + *b*^2 = *c*^2, where *a* and *b* are the two short sides and *c* is the hypotenuse. Remember that it only works on right triangles, but can be used to find the third side as long as two other sides are known.

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6th-8th Grade Math: Practice & Review55 chapters | 469 lessons

- The Pythagorean Theorem: Practice and Application 7:33
- The Pythagorean Theorem: Converse and Special Cases 5:02
- How to Find the Hypotenuse of a Right Triangle Using the Pythagorean Theorem 4:46
- How to Find Shorter Sides on a Right Triangle Using the Pythagorean Theorem 4:45
- How to Solve Functional Problems Involving the Pythagorean Theorem 5:52
- Go to 6th-8th Grade Geometry: The Pythagorean Theorem

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