Find the area of a square whose diagonal is six centimeters.
So here, we have a square where all angles are 90 degrees and all sides are congruent. So we are told that the diagonal which would connect two opposite corners say here and here. And we are asked to find the area. Well, connecting these corners, we are actually creating a right triangle two of them. However, only one is needed. It doesn’t matter which one. And notice that these sides which all the sides are congruent.
And in order to find the area, we need to know these sides because the area of a square is length times width or just the side squared because length and the width are the same. So we somehow need to find the length of 𝑥. Well, again, since we have this right triangle, we can use the Pythagorean theorem.
And the Pythagorean theorem states the square of the longest side is equal to the sum of the squares of the shorter sides. The longest side is the hypotenuse, the side across from the 90 degree angle. So we know that that is six. And then, the shorter sides are the legs, these ones, And they’re both 𝑥 and 𝑥.
And now, we can use this equation to solve for 𝑥. So six squared is 36 and 𝑥 squared plus 𝑥 squared is two 𝑥 squared. Now, to solve for 𝑥 squared, let’s get rid of two next to it. So we need to divide both sides of the equation by two. The twos cancels on the right and 36 divided by two is 18.
Now, before we square root both sides and solve for 𝑥, we set to find the area. We need to take the length times the width. But the length and the width were the same. They were both just 𝑥. So in order to find the area, we need to take 𝑥 times 𝑥. Well, 𝑥 times 𝑥 is 𝑥 squared. And we actually know what 𝑥 squared is equal to. It’s equal to 18. Therefore, our area is equal to 18 centimeters squared.
Now, let’s say that we didn’t recognize that the area is equal to 𝑥 squared. Say we kept solving for 𝑥, which means we would’ve square rooted both sides and found that 𝑥 is equal to the square root of 18. So when we would’ve plugged this in, we would have taken the square root of 18 times the square root of 18, which is the square root of 324 which is equal to 18. So once again, we would still get an area of 18 centimeters squared.