College Algebra: Using the Pythagorean Theorem

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A lot of times when one thinks about geometrical questions, one, or in this case me, we think about things that involve right triangles. And, you know, right triangles, of course, are things that throughout humanity have intrigued people and the most famous and probably the most important result about right triangles is the Pythagorean Theorem. The Pythagorean Theorem tells us that if you have a right triangle, then the length of the sides of the right triangle--by the way, besides that sort of hug up to the right angle, these are called legs. This triangle has a great pair of legs. Nothing beats a great pair of legs! Did you know that? I didn't make that up. That's a commercial, but I can't mention the product. This is called the hypotenuse. Well, it turns out that there's a relationship on a right triangle that links the length of the legs to the length of the hypotenuse.
And, in particular, if I call the length of one leg, let's say is a, and the length of the other leg is b, and the length of the hypotenuse is c, well then it turns out that the Pythagorean Theorem says that if you take a^2 and add it to b^2, what you get is c^2. So that's the Pythagorean Theorem and it's really, really valuable. In fact, this is actually used in real life. You know, carpenters actually use the Pythagorean Theorem all the time. And this is really true. This is not like fake real problems. If they want to make sure an angle is right if they're building like a deck or something and they want to make sure that angle is right, do you know what they do? What they do is they measure off 4 feet this way and they measure up 3 feet this way and they take those 2 points and then they put a string between them and they measure that length with a tape measure. And that has to be exactly 5 feet. And if it is, then they know they have a right angle because notice that 4, 3 and 5 are actually a right triangle because if you were to take 4^2, you get 16, 3^2 you get 9, 16 + 9 is actually 25 which notice, is 5^2. So, in fact, this is actually used all the time. And, in fact, what I just said is sort of the opposite of the Pythagorean Theorem, which is also true. Namely, if I had any triangle and it turns out that the square of 1 side plus the square of the other side equals the square of the 3^rd side, then that triangle had to be a right triangle to begin with. So the Pythagorean Theorem is really a fantastic fact. a^2 + b^2 = c^2. We use it all the time. And to illustrate that, I thought I would use it right now.
So as they say in show biz folks, "Take me out to the ball game." I just like hitting the--what is your title by the way? You're not a producer are you? You're a princess. That was Amy the Princess. I'll tell you about her some other time.
Let's consider a baseball diamond and you can see the reason why it's a baseball diamond is because it is shaped in the shape of a diamond. But, of course, if you look at it, if you have bad seats it might, in fact, look like a square which, of course, is what it really is. So this is a baseball on its side and you can see that, in fact, that this is a perfect square. I'll tell you it's a perfect square, and, in fact, here's a little ladybug. This is a superfluous lady bad running around the bases just to show you that, in fact, we can waste time on the web doing a lot of things such as the lady bug.
Now that we wasted the time with the ladybug, here is the profound question that I want to ask you. Here's a fact by the way. If you look at a regulation baseball diamond and measure it, you will see that each side is actually 90 feet. And the question is, in fact, there it is. You can probably write it already. How far is it from home plate, which is right here, to second base? So the base, of course, is home, 1st, 2^nd, and 3^rd, and pitchers mound. So how far is it? So what's that distance right there? And you notice that distance is actually part of a right triangle. It's this right triangle here. So what I want to do is I want to find the hypotenuse of this triangle. So I know that this length here is 90 feet and I know that this length right here is also 90 feet. And what I'm looking for is this length right here. And so that length is the hypotenuse. So if I think of that as h for hypotenuse, I know that the Pythagorean Theorem holds since this is a right triangle, this half of the field. And so what I know is that 1 leg squared, so 90^2 plus the other side which is also 90^2, that will equal the hypotenuse, which I'll call H^2. Now I want to solve this for H. So how would I do that? Well, first of all, I can add these 2 things up. Now you have to be a little careful here. Let's be really, really careful. If I have 1 90^2 and I add another 90^2, how many 90^2's do I have? I've got 2. So actually this equals 2 times 90^2. Don't be tempted to say sort of 2^2 or something. Don't bring the 2 in there. I just have 2 90^2 and that equals H^2. Now how would you solve this quadratic? Well, you know what I would do? Since I just have H^2 equals a number, I can just take plus or minus the square root of both sides. Something that I don't usually like to do, but in this case, oh, what the heck. It's just amongst us. Let's do it. So if we take plus or minus the square root of both sides, on this side I'll just have H. I'll write that here now. And on the other side I'd see the square root plus or minus 2 times 90^2. And what is that? Well, let's think about that for a second. First of all, should it be plus or minus. Which is the answer? Well, the answer is--gee, I think I moved this just a little bit too prematurely. Since H represents this length, that length has to be positive. So even though the mathematics produces 2 answers, given the actual application of this question, we know that, in fact, it's positive. So we can get rid of that other solution and we can say--why do I keep moving this? I keep moving this away. You'd think I hate baseball. Are you un-American? No, no, no. That's okay. Let's keep it here. So then I'd see that H would equal just the positive square root and the square root of a product is actually the product of the square root. It's one of the laws of exponents we've seen. So what I actually see is the square root of 2 times--and what's the square root of 90 squared? Well, you could either square out 90 and take the square root or realize the square root which is something to the one half power and the 90^2, those exponents sort of killed each other. Square roots kill squares. So I'm just left with the 90. So, in fact, I see an answer of 90. So that would seem like the answer. 90. That's the exact answer by the way. That is the exact length of this hypotenuse. The exact number of feet between home plate and second base. Are we done? Did we answer the question? No. Look at the question again. It says, "What is the distance to the nearest foot?" This is the exact answer, but they actually asked us for a rounded answer. So we have to take the square root of 2, take a calculator, compute the square root of 2, multiply it by 90 and what do we get? What we see is 127.27 stuff. But we're asked to round, so if we're going to round it, we'd see just 127 feet. So even though it's not the exact answer, the answer they wanted, which was a round answer, to the nearest foot it would be 127 feet. And look at that. I solved this problem without needed this space here even though I removed this three times. Amazing. Amazing.
And here's a little bonus question for you. Why do baseball players always scratch themselves? Well, the answer is they're thinking, `how can I be making so much money just throwing a little ball around and hitting it with the bat.'
All right, I'm sorry. Try some of the Pythagorean Theorem questions and have some fun.
Equations and Inequalities
Word Problems with Quadratics - Math Topics
Solving with the Pythagorean Theorem Page [2 of 2]