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# Pythagorean theorem with isosceles triangle

## Video transcript

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 are going to have two angles that are the same this angle is the same as that angle these 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 side this altitude of length 12 is on both triangles we know that both of these triangles are congruent so they're both going to have 13 they're gonna 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 2 and this is going to be x over 2 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 2 squared that's the base right over here this side right over here we could write that x over 2 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 into that same color this is going to be x squared over 4 that's just x squared over 2 squared plus 144 144 is equal to 13 squared is 169 now I can subtract 144 from both sides I'm going to 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 4 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 4 to isolate the x squared and so we get x squared is equal to 25 times 4 is equal to 100 now if you're just looking this purely mathematically you say oh 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 principal 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 5 so if we just look at this length right over here I'm doing that in the same color let me see so this length right over here that's going to be 5 and indeed 5 squared plus 12 squared that's 25 plus 144 is 169 13 squared so the key realization here is isosceles triangle the altitude splits it into two congruent right triangles and so it also splits this base into two so this is x over 2 and this is x over 2 and we use that information and the Pythagorean theorem to solve for x