Calculating Diagonals

Video Transcription

Hi. My name is Anthony Varela, and today I'd like to talk about calculating diagonals. So we're going to define what a diagonal is, and then we'll talk about how you go about calculating the diagonal length. And all of this involves squaring a number, and then taking the square root. So make sure you're familiar with those two steps.

So what is a diagonal? Well, let's imagine a soccer field. And the dimensions of this soccer field are 100 yards by 60 yards. And now imagine that there are two referees standing on opposite corners of this field.

What is the distance in between these two referees? Well, the line that I just drew is a diagonal to this rectangle. So what I know about a diagonal so far is that it's a straight line. I also know that it connects two opposite corners of this rectangle.

And this is actually true for all diagonals of rectangles or squares. You might also see diagonals in other polygons where the diagonal just connects two non-adjacent corners. But in this rectangle, they happen to be the opposite corners.

So now let's take a look at what happens when we slice the soccer field through using the diagonal line. And it actually creates two congruent triangles, or two triangles that are the same shape with the same size. And we can see that here I've drawn in these dashed lines to indicate that the opposite sides of this rectangle are equal in length, and then we have our 90-degree angles.

And so really what I have here is I have a triangle right here, and I have a congruent triangle-- or equal in size-- right here. So it cut the soccer field equally into two equal-sized pieces. So let's focus in on one of those triangles, and what we notice is that the diagonal of that rectangle has now become one of the sides to this triangle.

Now, what do I know about this triangle? Well, I know that it is a right triangle, because I see that 90-degree right angle right here. Now, the legs of this right triangle corresponds to the dimensions of that soccer field-- 100 yards and 60 yards. But then how can we talk about the diagonal of that rectangle, or this other side of this triangle?

We call that the hypotenuse. And that's a special name that we use to talk about this side in a right triangle. So we're going to write that down-- that this side of this right triangle's called the hypotenuse. And it's always the side that's opposite of the right angle.

So here's the right angle. Our two legs are adjacent to this right angle, and what's opposite to the right angle is the hypotenuse. So maybe you know something about right triangles and the hypotenuse of a right triangle, and that is that the hypotenuse can be described using the Pythagorean theorem.

So what is the Pythagorean theorem? Well, that says that a squared plus b squared equals c squared. So now let's talk about what a, b, and c are. Well, we can call the legs of the right triangle a and b, and then the hypotenuse is c.

So I'm calling 60 yards length a, 100 yards length b, and then our hypotenuse length c. And I should point out that it doesn't matter if I called this b or this a, as long as I'm always calling my hypotenuse c. So Pythagorean theorem says that the hypotenuse length squared equals the squared sums of the other legs in the right triangle.

So now let's go ahead and use, then, the Pythagorean theorem to calculate this diagonal, or this hypotenuse. So here's our picture again. I've just turned it a different way. And we see that a is 60 yards b is 100 yards, and then we have our hypotenuse, which is unknown. And we'd like to use the Pythagorean theorem to solve for c.

So what we can do is plug in what we already know, so that would be substitute 60 for a and 100 for b. So now my equation reads 60 squared plus 100 squared equals c squared. And now we need to square 60 and square 100.

Now, you can use your calculator if you're not sure, off the top of your head, what 60 squared is, or what 100 squared is. But I like to do this mentally. Just say, OK, well, 6 squared is 36, and then I need to add two 0's to that. So it's 3,600. And then 100 squared would be 100 times 100, or 10,000.

So now we just need to add these two numbers up, and that will tell me what c squared is. So 13,600 equals c squared. Well, 13,600 yards is not the length of the hypotenuse. It's not the diagonal of the soccer field. That is just way large in comparison to our other dimensions.

What 13,600 is is it's the hypotenuse squared. So we need to take the square root of both sides of this equation, because taking the square root of c squared would bring us back to c. So square root of 13,600-- you can use a calculator to evaluate this. And it's important to note this is now an approximation. So the diagonal of the soccer field is approximately 116.62 yards in length.

Now, I'd like to show you another way that you can calculate this hypotenuse length, or the diagonal of the soccer field. And you're going to get the same answer, just a different process. What you can do is manipulate this original equation-- the Pythagorean theorem, a squared plus b squared equals c squared-- and you can write it in terms of c alone-- because that's our main goal-- and then do substitution last.

So what we do, then, is we take the square root of both sides. That's how we're going to undo c being squared. So taking the square root of both sides. Now, it's very important to note that this does not simplify to a plus b equals c, because we need to take the square root of our entire quantity a squared plus b squared. So it's just simply the square root of the quantity a squared plus b squared equals c-- the hypotenuse to this right triangle, or remember, that's the diagonal of the soccer field.

OK, so we're going to write that down as another way to think about the Pythagorean theorem-- that the square root of a squared plus b squared equals c. So now we have something written in terms of c that's what we want to know, so now we can plug in our values for a and b and solve.

So putting in 60 and 100 for a and b, we have 60 squared plus 100 squared. We're going to take the square root of all of that, and that will tell us what c is. So notice we're back to where we were in our original method. We're taking the square root of 13,600-- which is approximately 116.62, and we're talking about yards.

All right, so let's review what we talked about today. We talked about diagonals. And in a rectangle or a square, it's a straight line that connects opposite corners. And we talked about a right triangle having a 90-degree right angle, and the side that is opposite of that angle we called the hypotenuse.

And the Pythagorean theorem relates the legs of a right triangle to its hypotenuse, and it says that a squared plus b squared equals c squared, where a and b are the legs and c is the hypotenuse. And we talked about a different way that we can write the Pythagorean theorem is taking the square root of both sides of this equation. So we can say the square root of a squared plus b squared equals c. So thanks for watching this tutorial on diagonals and how to calculate diagonal lengths. Hope to catch you next time.