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I promised you that I'd give you some more Pythagorean theorem problems so I will now give you some more Pythagorean theorem problems and once again this is all about practice let's say I had a triangle let's say it is well let me just draw my right triangle oh that's that's an ugly-looking right triangle let me draw another one there I just drew an equally ugly looking right triangle and if I were to tell you that this side is 7 this side is 6 and I want to figure out this side well we learned in the last presentation what which of these sides is the hypotenuse well here's the right angle so the side opposite the right angle is the hypotenuse so what we want to do is actually figure out the hypotenuse so we know that 6 squared plus 7 squared is equal to the hypotenuse squared and then the Pythagorean theorem they use C to represent the hypotenuse oh well you see here as well is equal to C squared so we'll say C equals the length of the hypotenuse and 36 plus 49 is equal to C squared what's 36 plus 49 is 70 it's 85 85 is equal to C squared or C is equal to the square root of 85 and this is the part that most people have trouble with is actually simplifying the radical so let's see the square root of 85 can i factor 85 so it's the product of a perfect square and another number let's see 85 is it divisible by 4 so it won't be difficult I 60 nor any of the multiples of 4 let's see if it's divisible but it's the 85 5 goes into 85 how many times it goes into it now that's not a perfect square either I don't think 85 can be factored further as a product of a perfect square and another number so you might correct me I might be wrong this might be good exercise for you to do later but as far as I can tell we have gotten our answer the answer here is the square root of 85 and if we actually wanted to estimate what that is well let's think about it the square root of 81 is 9 and the square root of 100 is 10 so it's someplace in between nine and 10 and it's probably a little bit closer to nine so it's nine point something something something and that's a good reality check that makes sense if the side is six the side of seven nine point something something something makes sense for the sign of that length let me give you another problem let's say that this is 10 and this is 3 what is this side first let's identify our hypotenuse well we have our right angle here so the side opposite the right angle is the hypotenuse and it's also the longest side so it's 10 so 10 squared is equal to the sum of the squares of the other two sides is equal to 3 squared and let's just call this a wicked opportunity plus a squared well this is 100 is equal to 9 plus a squared or a squared is equal to C 100 minus 9 a squared is equal to 91 so a is equal to the square root of 91 and I don't think that can be simplified further either C 3 doesn't go into it under is 91 a prime number I'm not sure as far as I know we're done with this problem let me do let me give you another problem and actually this time I'm going to include one extra step just to confuse you because I think you're getting this a little bit too easily let's say I have a triangle and once again we're dealing all with right triangles now I never never even attempt to use the Pythagorean theorem unless you know for a fact that it's a right triangle in this example we know that this is a right triangle if I were to tell you the length of this side is five and if I were to tell you that the this angle is 45 degrees can we figure out the other two sides of this triangle well we can't use the Pythagorean theorem directly because the Pythagorean theorem tells us that if we have a right triangle and we know two of the sides that we can figure out the third side here we have a right triangle and we only know one of the sides so we can't figure out the other two just yet but maybe we could use this extra information right here this 45 degrees to figure out another side and then we'd be able to use the Pythagorean theorem well we know that the angles in a triangle add up to 180 degrees oh hopefully you know that the angles in a triangle add up to 180 degrees if you don't it's my fault because I haven't taught you that already so let's figure out what what what the angles of this triangle add up to well I mean we know they add up to 180 but using that information we could figure out what this angle is right because we know this angle is 90 this angle is 45 so we say 45 let's call this angle X I'm trying to make it messy okay 45 plus 90 all right this just implies this is a 90-degree angle plus X is equal to 180 degrees and that's because the angles in a triangle always add up to 180 degrees so if we just solve for X we get here we get 135 135 that's a 5 plus X is equal to 180 subtract 135 from both sides we get X is equal to 45 degrees interesting X is also 45 degrees so we have a 90 degree angle and two 45 degree angles now I'm going to give you another theorem that's not named after the religion of a religion or the founder of religion I actually don't think this this theorem has a name at all but it's the it's the fact that if I have a triangle to draw another triangle out here we're two of the base angles are the same and when I say a base angle I just mean two angles if these two angles are the same let's call it a they're both a then the sides that they don't share right these angles share this side right but if we look at the sides that they don't share we know that these sides are equal I forgot what we call this in geometry class maybe I'll look it up in another presentation Allah I'll let you know but I got this far without knowing what it what the name of the theorem is but just and it makes sense you almost you don't even need me to tell you that how could I draw this angle if if I were to change one of these angles the length would also change or another way to think about it the only way I don't want to confuse you too much but you can visually see that if these two sides are the same then these two angles are going to be the same if you changed one of these sides lengths then the angles will also change or the angles will not be equal anymore but I'll leave you that I'll leave that for you to think about but just take my word for it right now that if two angles and a triangle are equivalent then the sides that they don't share are also equal in length right so make sure you remember not the side that they share because I can't be equal to anything it's the side that they don't share is equal are equal in length so here we have an example where we have two equal angles they're both 45 degrees so that means that the sides that they don't share this is a side they share right both angles share this side so that means that the side that they don't share are equal so this side is equal to this side and I think you might be experiencing an aha moment right now well if this side is equal to this side we already gave you at the beginning this problem that this side is equal to five so then we know that this side is equal to five and now we can do the Pythagorean theorem we know this is the hypotenuse right hypotenuse so we could say five squared plus five squared is equal to let's go say C squared where C is the length of the hypotenuse well five squared plus five squared that's just fifty is equal to C square and then we get C is equal to the square root of 50 and 50 is 2 times 25 so C is equal to 5 square root of 2 interesting so I think I might have given you a lot of information there if you get confused maybe you want to re-watch this video but on the next video I'm actually going to give you more information about this type of triangle which is actually a very kind of common type of triangle you'll see in geometry and trigonometry called a 45-45-90 triangle and it makes sense why it's called that because it has 45 degrees 45 degrees and a 90 degree angle and I'll actually show you a quick way of using that information that it is a 45 45 90 degree triangle to figure out the sides if you're given even one of the sides I hope I will confuse you too much and I look forward to seeing you in the next presentation see you later