Int Algebra: The Pythagorean Theorem

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Quadratic Equations and Inequalities
Formulas and Applications
The Pythagorean Theorem: Another Example [1 of 1]
Hey I am so excited because I get to share with you one of my all time favorite theorems of all the mathematics, mathematics is huge, lots of theorems and this is one of my all time favorites. It’s one of the most important theorems on mathematics it is of course the Pythagorean Theorem. And you know what it says, it says that it hit a right triangle and you know the lengths of the sides. If one leg is A another leg is B and you know the hypotenuse is C then those lengths magically work together in such a way that A2 when added to B2 will always equals C2 that’s right. A2+B2=C2 do you love it sing with me that’s okay. Alright anyway, let me show you some wonderful applications where we can use this important formula whenever we have a right triangle, where we know something about some of the lengths of the size. For example, check it out, check it out, check it out so here’s a right triangle we know that by the way because that little tiny right angle symbol there. And I am told that the hypotenuse has length 20 centimeters. I am told that this leg has length 12 centimeters and now I want to find out this missing length. The missing length here is easy to figure out because of course Pythagorean Theorem comes to our rescue. So, what do I know? I know that X2 that length squared, plus this leg squared, which is 122 is going to equal the hypotenuse squared or X2+144 equals and then 202 is 400. If I subtract 144 from both sides I see that X2 equals 256. Well that’s awesome, because I can now just take the square root of both sides. Now, technically just to remind you, you are going to be taking plus or minus the square root, but X represents a length and so since X represents a length we know it’s going to be positive, pick up a ruler, look at it there is no negative numbers on there, it’s all positive and so therefore you see the positive square root, X equals the positive square root of 256, which equals 16 and so X=16 cm and you see how we found that length by actually using the Pythagorean Theorem which we can recall. Let me show you in a more abstract question. So, let’s raise the level up, you know prove a little bit more crazy, little more abstract. I want us to figure out what X is, if all we know is that we have a right triangle, one length of one leg is 8, the other leg has length X and hypotenuse has length X+2, we have no idea what X is, your mission should you decide to accept it is to figure out what X is. So now how we do it? Well, what we do is use the Pythagorean Theorem, which says that this leg squared which is X2 plus this leg squared which is 82 has to equal this leg squared. And I am going to make a mistake right here, let’s see if you can catch my mistake, here’s the mistake. What's the mistake? The mistake is that this as written is only squaring the 2, this is technically (X+2)2, which is X+4 that’s wrong, you have to square this entire value which means that we have to put parenthesis around the whole thing. Now, if that didn’t stump you awesome, congratulations I love you. If it did stump you, I still love you but I want you to really remember that in fact we have to put parenthesis around quantities, when we are squaring the entire quantity. Isn't that great lesson? Thank you very much and we are done. Okay, now if we work this out, I see X2+(82) is 64 and what does that equal. Well now I have to actually foil and multiply (X+2) by (X+2), well X times X is X2 the outside terms give me a 2X and the inside terms give me a 2X, so when I add them together I get a 4X and then 2 times 2 is of course 4 and so I am left with this. I have noticed this is actually kind of cool because if I subtract X2 from both sides, this seemingly quadratic equation magically becomes a linear equation, check it out. X2-X2=0 I am just left with 64 equals X2-X2=0 I am left with 4X+4, so now I subtract 4 from both sides, subtract 4 from both sides and I am left with 60=4X+0. Now to solve for X I just is divide by 4 and 60 divided by 4 is equal to 15. And so X=15, really kind of cool actually, if you think about it because if we have you know quadratic equation, there might be two solutions, but that wouldn’t make any sense because there is only one triangle you can see this one triangle here, so that will be just one answer and the cool thing is that the math won't permit anything else and in fact their potential square terms that might generate two solutions dropout and we are only left with 1 sole solution X=15 units that’s the length of this leg and we can actually find the length of hypotenuse is going to be 15+2, AKA 17, see you soon.