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### Course: Geometry 227-228 > Unit 1

Lesson 3: Pythagorean theorem- Intro to the Pythagorean theorem
- Pythagorean theorem example
- Pythagorean theorem intro problems
- Use Pythagorean theorem to find right triangle side lengths
- Pythagorean theorem with isosceles triangle
- Use Pythagorean theorem to find isosceles triangle side lengths
- Right triangle side lengths
- Use area of squares to visualize Pythagorean theorem

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# Pythagorean theorem with isosceles triangle

To find the value of a base (x) in an isosceles triangle, first split the triangle into two congruent right triangles by drawing an altitude. Then, use the Pythagorean theorem to create an equation involving x. Finally, solve the equation to find the unknown base, x.

## Want to join the conversation?

- In1:09, why would the base of the isosceles triangle be x/2?(57 votes)
- The base of the isosceles triangle is not x/2, it is x. Whereas x/2 is the base of each of the right triangles.(82 votes)

- What if we don't have the height available?(17 votes)
- It won't be easy but if you look carefully at the isosceles triangle it's a 45, 45, 90 triangle when split in half

And to find the hypotenuse you have to multiply by the square root of 2 but we are not trying to find the hypotenuse we are trying to find the height

So we have to do the opposite instead of multiplying by the square root of 2 you have to divide by the square root of 2

So we already know the hypotenuse which is 13 so it would beusually we can leave it like this but we can also rationalize it by multiplying`(13/√2)`

which is approximately 9.19`(13/√2)`

with`(√2/√2)`

Hopefully you found that helpful :)

(And in case you are wondering why the height is not the same is because the drawing in the video is not up to scale if the hypotenuse is 13 then really if you want to be exact then 9.19 is probably your best bet but now you should just roll with it)(32 votes)

- Why at0:17he said that these are right angles? There's nothing in the diagram that says so.(14 votes)
- The altitude that is dropped is perpendicular to the base of the triangle. Altitudes are dropped from the vertex of the triangle and intersect with the base of the triangle to form right angles. I hope this helps you.(22 votes)

- Why did 2 as the denominator become 4?(12 votes)
- Because (x/2)^2 = (x/2)(x/2) = (x*x)/(2*2) = x^2/4

Hope this helps.(21 votes)

- Can't you just take the 25 and square root that to find the answer instead of taking x/4^2 . (x over 4 squared)??(11 votes)
- Yes, you can take the square root of 25 to get 5, but realize that 5 is only half the value of x. Double the 5 to get x = 10.(5 votes)

- Why is that sometimes when your solving a+b=c you subtract or add, I don't get that part, sometimes you have to add and sometimes you have to subtract to get the right answer, how do you know for sure which one to do? When your solving the Pythagorean Theorem(4 votes)
- Anything you do in Pythagorean Theorem (PT) is the opposite of what it is. Say you have a negative, you have to add that to the other side of you use it, you cant divide or subtract or multiply. You have to subtract.(0 votes)

- How do I find X if I don't have something down the middle? I only have 2 sides and a 90-degree and 58-degree angle?(7 votes)
- Hi I hope this is helpful, if you divide it down the center and find the base length of both sides then add them together you should get it.(4 votes)

- In the case of this particular problem only, couldn't you do 13^2-12^2, and then get 25 and square root it, then do 5x2 and then get the answer? I was just wondering if this problem-solving technique was realistic for these kinds of problems or if we should just stick to what Sal was using. Thanks.(4 votes)
- Yes, the problem-solving technique you mentioned works great! You can use this technique any time you are given two equal sides of an isosceles triangle and the altitude to the third side, and you are asked to find the third side.

Have a blessed, wonderful day!(7 votes)

- This is a great and interesting way to do this. The way I originally found the answer was to compute regularly with the Pythagora's theorem and then multiply the final result by 2 to give the actual whole length of x. The x/2 principle will also work.(5 votes)
- I just do this: if x and y are not the Hypotenuse and z is the Hypotenuse (also a stands for answer) then I do x^2+y^2=a. then the square root of a is the Hypotenuse or z. if you are trying to find a side other than (lets say y) the Hypotenuse I do z^2-x^2=a

then I find the square root of a and that will be equal to y.(2 votes)

- What would you do with it is on the hypotenuse(2 votes)
- Hi! I am not really sure, what you mean but if x was the the hypotenuse, then you'd divide the base by 2, carry out Pythagorean Theorem which would give you the answer to the hypotenuse!

Hope this helped! :)(4 votes)

## Video transcript

- [Instructor] We're asked
to find the value of x in the isosceles triangle shown below. So that is the base of this triangle. So pause this video and see
if you can figure that out. Well the key realization to solve this is to realize that this
altitude that they dropped, this is going to form a right angle here and a right angle here and notice, both of these triangles, because this whole thing
is an isosceles triangle, we're going to have two
angles that are the same. This angle, is the same as that angle. Because it's an isosceles triangle, this 90 degrees is the
same as that 90 degrees. And so the third angle
needs to be the same. So that is going to be the same as that right over there. And since you have two
angles that are the same and you have a side between them that is the same this altitude of 12 is on both triangles, we know that both of these
triangles are congruent. So they're both going to have 13 they're going to have one side that's 13, one side that is 12 and so this and this side are going to be the same. So this is going to be x over two and this is going to be x over two. And so now we can use that information and the fact and the Pythagorean Theorem to solve for x. Let's use the Pythagorean Theorem on this right triangle on the right hand side. We can say that x over two squared that's the base right over here this side right over here. We can write that x over two squared plus the other side plus 12 squared is going to be equal to
our hypotenuse squared. Is going to be equal to 13 squared. This is just the Pythagorean Theorem now. And so we can simplify. This is going to be x. We'll give that the same color. This is going to be x squared over four. That's just x squared over two squared plus 144 144 is equal to 13 squared is 169. Now I can subtract 144 from both sides. I'm gonna try to solve for x. That's the whole goal here. So subtracting 144 from both sides and what do we get? On the left hand side, we have x squared over four is equal to 169 minus 144. Let's see, 69 minus 44 is 25. So this is going to be equal to 25. We can multiply both sides by four to isolate the x squared. And so we get x squared
is equal to 25 times four is equal to 100. Now, if you're just looking
this purely mathematically and say, x could be
positive or negative 10. But since we're dealing with distances, we know that we want the
positive value of it. So x is equal to the principle root of 100 which is equal to positive 10. So there you have it. We have solved for x. This distance right here, the whole thing, the whole thing is
going to be equal to 10. Half of that is going to be five. So if we just looked at
this length over here. I'm doing that in the
same column, let me see. So this length right over here, that's going to be five and indeed, five squared plus 12 squared, that's 25 plus 144 is 169, 13 squared. So the key of realization here is isosceles triangle, the altitudes splits it into
two congruent right triangles and so it also splits this base into two. So this is x over two and this is x over two. And we use that information and the Pythagorean Theorem to solve for x.