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## Basic geometry and measurement

### Unit 13: Lesson 3

Pythagorean theorem application- Use Pythagorean theorem to find area of an isosceles triangle
- Use Pythagorean theorem to find perimeter
- Use Pythagorean theorem to find area
- Pythagorean theorem word problem: carpet
- Pythagorean theorem word problem: fishing boat
- Pythagorean theorem word problems
- Pythagorean theorem in 3D
- Pythagorean theorem in 3D

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# Use Pythagorean theorem to find area of an isosceles triangle

Sal uses the Pythagorean theorem to find area of an isosceles triangle.

## Want to join the conversation?

- Can you tell me what is the use of Pythagorean in our daily

life?(37 votes)- Here is an answer from Hadriel Ramos (check the comments!) :Actually it is one of the most fundamental theorems in basic geometry.

Just imagine a tall post (10 meters High) and you are standing 20 meters away. And you wanted to tie a rope to the high point of the post to the ground from which you are standing. You use the Pythagorean Theorem.(18 votes)

- Have a question for right triangle. Need help to solve it.

The hypotenuse of right triangle is 26cm. The sum of two other sides is 34cm. Find the length of two sides.(16 votes)- Well, we'll start by defining the sides. one unknown side will be x, and one will be 34 - x. Now, the Pythagorean theorem says that x^2 + (34 - x)^2 = 26^2.

Multiplying out, we get:

x^2 + 34^2 + x^2 - 68x = 676

Simplifying:

2x^2 - 68x + 1156 = 676

2x^2 - 68x + 480 = 0

x^2 - 34x + 240 = 0

Now, we have a quadratic equation, which we can factor:

{x^2 - 34x + 240 = 0} = {(x - 24)(x - 10) = 0}

Now we can solve for x by dividing by x - 10:

(x - 24)(x - 10) = 0

x - 24 = 0

And by dividing by x - 24:

x - 10 = 0

So, we have that x = 10, and x = 24. This means that the answer to your problem is [24, 10].(33 votes)

- Can you use the Pythagorean Theorem to solve for any triangle? If so, can you please add an example for a scalene triangle.(6 votes)
- You can work out the area for triangles which are made up with right triangles, like the triangle that Sal uses in this video.

Hope this helps.(7 votes)

- How if the value of the base is missing(12 votes)
- If the value of the base is missing , than you dont use pythagorean theroem for it.(5 votes)

- Is it possible to create a video for using the Pythagorean theorem to find the perimeter?(6 votes)
- At3:00, Sal said 100 and 44, not 144. I thought you only use "and" when you are referring to decimals?(4 votes)
- lots of people say 100 and 44 but they mean 144 its probably because of his accent but yes you are really only supposed to use and when you are referring to decimals does that help answer your question?(4 votes)

- m

2 = x + 91 how do i find the solution to this eqation with an isosyles triangle that has a degree of 50(5 votes) - why are we learning this?(4 votes)
- Every skill that we learn, no matter how insignificant it may seem, will be used someday. Most, if not all, of the lessons here on Khan Academy can help with at least one job where it would be vital to have an understanding of it.(3 votes)

- Is it just me, or is Pythagorean Theorem ALMOST as fun as Minecraft (or Minekahn, for that matter) and fractions?(4 votes)
- i cant describe my loathing for the pethagriam(4 votes)

## Video transcript

- [Tutor] Pause this video
and see if you can find the area of this triangle, and I'll give you two hints. Recognize, this is an isosceles triangle, and another hint is that
the Pythagorean Theorem might be useful. Alright, now let's work
through this together. So, we might all remember
that the area of a triangle is equal to one half times
our base times our height. They give us our base. Our base right over here is, our base is 10. But what is our height? Our height would be, let me do this in another color, our height would be the length
of this line right over here. So, if we can figure that out, then we can calculate what
one half times the base 10 times the height is. But how do we figure out this height? Well, this is where
it's useful to recognize that this is an isosceles triangle. An isosceles triangle has
two sides that are the same. And so, these base angles are
also going to be congruent. And so, and if we drop an
altitude right over here which is the whole
point, that's the height, we know that this is, these
are going to be right angles. And so, if we have two triangles where two of the angles are the same, we know that the third angle
is going to be the same. So, that is going to be congruent to that. And so, if you have two triangles, and this might be obvious
already to you intuitively, where look, I have two angles in common and the side in between them is common, it's the same length, well that means that these are going to be congruent triangles. Now, what's useful about
that is if we recognize that these are congruent triangles, notice that they both have a side 13, they both have a side, whatever
this length in blue is. And then, they're both
going to have a side length that's half of this 10. So, this is going to be five,
and this is going to be five. How was I able to deduce that? You might just say, oh that
feels intuitively right. I was a little bit more rigorous here, where I said these are
two congruent triangles, then we're going to split this 10 in half because this is going to be equal to that and they add up to 10. Alright, now we can use
the Pythagorean Theorem to figure out the length of
this blue side or the height. If we call this h, the
Pythagorean Theorem tells us that h squared plus five
squared is equal to 13 squared. H squared plus five squared, plus five squared is going
to be equal to 13 squared, is going to be equal to our longest side, our hypotenuse squared. And so, let's see. Five squared is 25. 13 squared is 169. We can subtract 25 from both sides to isolate the h squared. So, let's do that. And what are we left with? We are left with h squared is equal to these canceled out, 169 minus 25 is 144. Now, if you're doing it
purely mathematically, you say, oh h could be plus or minus 12, but we're dealing with the distance, so we'll focus on the positive. So, h is going to be equal
to the principal root of 144. So, h is equal to 12. Now, we aren't done. Remember, they don't want us to just figure out the height here, they want us to figure out the area. Area is one half base times height. Well, we already figured out that our base is this 10 right over here, let me do this in another color. So, our base is that distance which is 10, and now we know our height. Our height is 12. So, now we just have to compute
one half times 10 times 12. Well, that's just going to be equal to one half times 10 is five, times 12 is 60, 60 square units, whatever
our units happen to be. That is our area.