How to Find Shorter Sides on a Right Triangle Using the Pythagorean Theorem

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  • 0:01 Pythagorean Theorem
  • 0:53 Altering the Formula
  • 1:43 Pythagorean Triangles
  • 2:31 Other Right Triangles
  • 3:46 Lesson Summary
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Instructor: Kevin Newton

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

Sure, the Pythagorean Theorem is useful for finding the hypotenuse of a right triangle, but what about the other sides? As this lesson shows, it's just a matter of flipping the formula.

Pythagorean Theorem

Perhaps no Ancient Greek has meant as much to would-be geometry students than Pythagoras. The Pythagorean Theorem, which is the formula of A^2 + B^2 = C^2 for right triangles, may be the first time that math looks simple and elegant. If you're lucky, you had a teacher who showed the truth of this with a nice animation comprised of a triangle surrounded by 3 squares. If not, here's one to show you how it works!

pythagorean theorem

In any event, if you drag out the right triangle to become bigger or smaller, the squares shrink as well. At any point, you can drag the two smaller squares into the bigger square and see that the area is equal. Sure, you could memorize the formula and it's not a hard one to learn. Still, seeing it in action is a great way to appreciate just how smart the whole thing really is.

Altering the Formula

Of course, finding the hypotenuse, the longest side of a triangle that is most often called C, is not always what we're trying to do. Sometimes we have the hypotenuse and need to figure one of the other sides. Luckily, that's still pretty easy to do. Simply enter all the information that you know into the formula.

Let's say you had a triangle with a known side of 8 and a hypotenuse of 10. So what do you do? Do you plug the numbers in as A^2 + 8^2 = 10^2? No! You can do better than that! 8^2 is 64 and 10^2 is 100. Subtract 64 from 100, leaving only the A^2 on that side of the equal sign. That means A^2 is equal to 36. From here, you can just take the square root and find that A is equal to 6.

Pythagorean Triangles

Sometimes, you may wonder how your math teachers are always able to come up with such clear example problems, especially when dealing with square roots. However, when it comes to the Pythagorean Theorem problems, they have a secret weapon known as Pythagorean triple. These triangles have sides that are multiples of 3, 4, and 5, respectively. The math always works pretty cleanly. If you really wanted to look smart, you could multiply 3, 4, and 5 by 11, 111, or even 1,111 and still get the same effect. So, if you want to look really smart, go ahead and do the numbers for a triangle with the sides 333, 444, and 555. It'll work out, trust me.

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