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Use Pythagorean theorem to find area of an isosceles triangle

Video transcript

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 all right now let's work through this together so we might all remember that the area of a triangle is equal to 1/2 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 1/2 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 and 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 or 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 they both have a side 13 they both have a side whatever this length blue is and so and then they're both going to have a side length that's half of this 10 so this is going to be 5 and this is going to be 5 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 ideas 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 all right now we can use the Pythagorean theorem to figure out the length of the blue side or the height if we call this H the Pythagorean theorem tells us that H squared plus 5 squared is equal to 13 squared h squared plus 5 squared plus 5 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 5 squared is 25 13 squared is 169 we can subtract 25 from both sides to isolate the eight squares so let's do that and what are we left with we are left with H squared is equal to these cancelled out 169 minus 25 is 144 now if you're doing it purely mathematically coh 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 to just figure out the height here they want us to figure out the area area is 1/2 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 have to compute 1/2 times 10 times 12 well that's just going to be equal to 1/2 times 10 is 5 times 12 is 60 60 square units whatever our units happen to be that is our area