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### Course: High school geometry>Unit 5

Lesson 1: Pythagorean theorem

# 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?

• In , why would the base of the isosceles triangle be x/2?
• 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.
• What if we don't have the height available?
• 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 be (13/√2) usually we can leave it like this but we can also rationalize it by multiplying (13/√2) with (√2/√2) which is approximately 9.19

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)
• Why at he said that these are right angles? There's nothing in the diagram that says so.
• 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.
• Why did 2 as the denominator become 4?
• Because (x/2)^2 = (x/2)(x/2) = (x*x)/(2*2) = x^2/4
Hope this helps.
• 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)??
• 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.
• 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
• 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.
• 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?
• 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.
• 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.
• 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!
• 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.