Video Transcript
What is the area of this square?
Well, the only bit of information we’ve been given about the square is the fact that its diagonal is five centimeters long. But how is this going to help us? Well, what this diagonal has effectively done is divided our square into two triangles. And they’re two identical triangles. But what else is special about these triangles? Well, because it’s a square, both these triangles are right triangles. Because if we take a look, we can see that the angle at 𝐵 and the angle at 𝐷 are both going to be right angles within our triangles.
Well, we’ve said that we’ve got right triangles, but why is this useful? Well, if we’re trying to find the area of the square, an area of a square is equal to the side length squared. Well, therefore, we need to calculate what one of the side lengths are. And how are we going to do this? Well, we’re going to do this using the fact that we have a right triangle.
Well, what we’re gonna concentrate is on the triangle 𝐴𝐵𝐶. It doesn’t really matter; we could’ve used the one at the top side. Now, if we’ve got this triangle, we can see that both of the side lengths that aren’t the hypotenuse, which is the diagonal, are going to be the same length because it’s a square. So, we’ve got 𝑠 and 𝑠 is what we’re gonna call these two sides.
So, to help us find our side length, what we’re gonna use is the Pythagorean theorem. And this tells us that 𝑐 squared equals 𝑎 squared plus 𝑏 squared, where 𝑐 is the hypotenuse or longest side. And in a right triangle, the hypotenuse or longest side is always opposite the right angle. Well, to enable us to find out what our side lengths are in the triangle that we’ve got here, what we’re gonna do is substitute the values into the Pythagorean theorem.
Well, when we do that, what we’re gonna have is five squared is equal to 𝑠 squared plus 𝑠 squared. Okay, can we simplify this a bit? Well, yes, we can because if we simplify this, what we’re gonna get is 25, and that’s cause five multiplied by five is 25, is equal to two 𝑠 squared. And that’s because if we got one 𝑠 squared and we had another one, that gives us two 𝑠 squared.
Well then, if we’ve got two 𝑠 squared and we want to find one 𝑠 squared, what we’re gonna do is divide both sides of the equation by two. So we get 12.5 is equal to 𝑠 squared. Well, if we wanted to, what we could do now is take the square root of both sides of our equation to find out what 𝑠 is, so to find out what one of our side lengths is. And this would tell us that each of our side lengths are root 12.5. And we could calculate what this was. However, we don’t need to do this step.
But why? Well, we don’t need to do this step because, as we said at the beginning, the area is equal to the side length squared or the area is equal to 𝑠 squared. Well, in this line here, we have the value of 𝑠 squared. So therefore, we can surmise that the area is going to be equal to 12.5. And then, if we remember the units, it’s going to be centimeters squared.
