Main content

### Course: Geometry (all content) > Unit 9

Lesson 1: Pythagorean theorem- Intro to the Pythagorean theorem
- Pythagorean theorem example
- Pythagorean theorem intro problems
- Use Pythagorean theorem to find right triangle side lengths
- Use Pythagorean theorem to find isosceles triangle side lengths
- Right triangle side lengths
- Use area of squares to visualize Pythagorean theorem

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Pythagorean theorem example

The Pythagorean theorem states that in a right triangle, the sum of the squares of the two shorter sides equals the square of the longest side (the hypotenuse). We can apply the theorem to find the missing side length of a right triangle, even when the missing length is one of the shorter sides. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- 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 )(22 votes)- wow reading this... it says 11 years ago(10 votes)

- Why didn't Sal just go all the way and square root 115? Are you not supposed to have a decimal?(6 votes)
- 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.(31 votes)

- i really dont get y we need this like i am good without this in my life(11 votes)
- This is a fundamental part of architecture, buildings, etcetera. If you don't like this, then you'll never be able to buy a proper house and be easily scammed.(8 votes)

- How do you calculate a square root without a calculator ? Is it possible ?(8 votes)
- Daphne78, it is possible. For example, (-5)*(-5)=25, also 5*5=25. So √25 can either be (-5) or 5.(2 votes)

- So is the Pythagorean theorem for other triangles too, or just for the right triangles?(2 votes)
- 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!(12 votes)

- 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

?(5 votes)- Square root of 115 equals 10.723..., not 23. Because square root of 115 is not a whole number, we just keep it written as √115.(6 votes)

- Is the pythagorean theorem only valid for right triangles because I am only seeing examples done on right triangles.(3 votes)
- It only works with right triangles as Sal says at0:59(5 votes)

- But can you solve the question if i just give you the hypotenuse and another side to solve the third side?(3 votes)
- Yes. You can use the Pythagorean theorem to solve for a missing side.(4 votes)

- if we had a left 90o degree triangle, would the Pythagorean theorem work????(0 votes)
- The Pythagorean theorem ONLY works on 90 degree triangle aka right triangle.(3 votes)

- Why didn’t he just go All the way to square root your supposed to have a decimal.(3 votes)
- 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.(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.