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Pythagorean theorem example

Sal uses the Pythagorean theorem to find the height of a right triangle with a base of 9 and a hypotenuse of 14. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • blobby green style avatar for user Leslie Caro Orozco
    Hey! At you say that if we know two of the sides we can use the pythagorean theorem, what can we do or what can we use if we only know one side?
    (52 votes)
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    • old spice man green style avatar for user Skywalker94
      The Pythagorean theorem only works if you know two sides. If you only know one side and the triangle has been drawn accurately to scale, you might be able to get away with using a protractor and a ruler, but that again relies that you have the triangle actually drawn out and that it is to scale. I wouldn't advise it if accuracy is a concern unless you are actually instructed to do so.
      (61 votes)
  • blobby green style avatar for user lovebori
    I was wondering can i just solve the problem like this??
    a 2+9 2=14 2
    so 14 -9 = 5
    so can't it be 5 squared???( 5 2 )
    (5 votes)
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  • blobby green style avatar for user andon bizzell
    i really dont get y we need this like i am good without this in my life
    (10 votes)
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  • leafers ultimate style avatar for user hunt885
    Why didn't Sal just go all the way and square root 115? Are you not supposed to have a decimal?
    (6 votes)
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    • piceratops ultimate style avatar for user Robert Johnston
      That’s a good question. Most of what is taught here is what we call theoretical math. The convention for this, is to express an answer in the simplest format. To express an answer as root 115 is a lot cleaner and actually more accurate than the decimal form (10.72380529…). Now, if you need to actually apply this number to a physical object; say you need to measure the length of something, the decimal format would better suit this.
      (23 votes)
  • leaf blue style avatar for user daphne78
    How do you calculate a square root without a calculator ? Is it possible ?
    (8 votes)
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  • blobby green style avatar for user Aaron waterboy
    So when he says Square root of 115 he means 23 as the answer or just 115 why did he say 115 is the answer? Is there a rule behind it, if the numbers square rooted from the original number are prime then the original number is the answer
    ?
    (6 votes)
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  • male robot johnny style avatar for user Isaiah Walker
    So is the Pythagorean theorem for other triangles too, or just for the right triangles?
    (3 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      The Pythagorean theorem applies to right triangles only.

      It is true that some problems involving non-right triangles can be solved using the Pythagorean theorem indirectly, but only by creating right triangles. For example, given the sides of an isosceles triangle, one can find the altitude to the unequal side by drawing this altitude, and then using the Pythagorean theorem on one of the two congruent right triangles formed by this altitude.

      Have a blessed, wonderful day!
      (11 votes)
  • leaf blue style avatar for user maxsisco
    if we had a left 90o degree triangle, would the Pythagorean theorem work????
    (0 votes)
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  • blobby green style avatar for user s23wili9702
    Why didn’t he just go All the way to square root your supposed to have a decimal.
    (4 votes)
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    • leaf green style avatar for user Nikolay
      You know about irrational numbers where the decimal goes on forever?

      Which would you rather see? Sqrt(15) or 10.7238052948? Imaging performing operations on such a long decimal!

      Decimals are not always preferred in maths (especially in algebra) for this reason. It's preferred that you leave it as a root or fraction to make the working easier.
      (5 votes)
  • starky sapling style avatar for user TayEmery22
    Do we have to simply the square root all the way down? I was told to just divide it on my calculator, and I was not really taught rationales and how to simply them
    (3 votes)
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    • duskpin sapling style avatar for user Lin Gh.
      Well, in my opinion, letting students depend on the calculator instead of trying to find the solution by themselves, is one of the biggest mistakes in the highschool curriculum. (I don't know about the elementary school) I've met some people who depend on the calculator all the time that they don't even remember the multiplication table.
      So, I highly advice you, to familiarize yourself with simplifying equations and numbers, and try to avoid the calculator as much as possible.

      But, to answer your question, yes. You don't have to, but what's the point of math if you're letting the calculator do all the work?
      (4 votes)

Video transcript

Say we have a right triangle. Let me draw my right triangle just like that. This is a right triangle. This is the 90 degree angle right here. And we're told that this side's length right here is 14. This side's length right over here is 9. And we're told that this side is a. And we need to find the length of a. So as I mentioned already, this is a right triangle. And we know that if we have a right triangle, if we know two of the sides, we can always figure out a third side using the Pythagorean theorem. And what the Pythagorean theorem tells us is that the sum of the squares of the shorter sides is going to be equal to the square of the longer side, or the square of the hypotenuse. And if you're not sure about that, you're probably thinking, hey Sal, how do I know that a is shorter than this side over here? How do I know it's not 15 or 16? And the way to tell is that the longest side in a right triangle, and this only applies to a right triangle, is the side opposite the 90 degree angle. And in this case, 14 is opposite the 90 degrees. This 90 degree angle kind of opens into this longest side. The side that we call the hypotenuse. So now that we know that that's the longest side, let me color code it. So this is the longest side. This is one of the shorter sides. And this is the other of the shorter sides. The Pythagorean theorem tells us that the sum of the squares of the shorter sides, so a squared plus 9 squared is going to be equal to 14 squared. And it's really important that you realize that it's not 9 squared plus 14 squared is going to be equal to a squared. a squared is one of the shorter sides. The sum of the squares of these two sides are going to be equal to 14 squared, the hypotenuse squared. And from here, we just have to solve for a. So we get a squared plus 81 is equal to 14 squared. In case we don't know what that is, let's just multiply it out. 14 times 14. 4 times 4 is 16. 4 times 1 is 4 plus 1 is 5. Take a 0 there. 1 times 4 is 4. 1 times 1 is 1. 6 plus 0 is 6. 5 plus 4 is 9, bring down the 1. It's 196. So a squared plus 81 is equal to 14 squared, which is 196. Then we could subtract 81 from both sides of this equation. On the left-hand side, we're going to be left with just the a squared. These two guys cancel out, the whole point of subtracting 81. So we're left with a squared is equal to 196 minus 81. What is that? If you just subtract 1, it's 195. If you subtract 80, it would be 115 if I'm doing that right. And then to solve for a, we just take the square root of both sides, the principal square root, the positive square root of both sides of this equation. So let's do that. Because we're dealing with distances, you can't have a negative square root, or a negative distance here. And we get a is equal to the square root of 115. Let's see if we can break down 115 any further. So let's see. It's clearly divisible by 5. If you factor it out, it's 5, and then 5 goes in the 115 23 times. So both of these are prime numbers. So we're done. So you actually can't factor this anymore. So a is just going to be equal to the square root of 115. Now if you want to get a sense of roughly how large the square root of 115 is, if you think about it, the square root of 100 is equal to 10. And the square root of 121 is equal to 11. So this value right here is going to be someplace in between 10 and 11, which makes sense if you think about it visually.