The area of the square 𝐴𝐵𝐶𝐷 is nine over two centimeters squared. Find the length of line 𝐵𝐷.
Almost immediately, we probably think the area of a square formula area equals side squared, which means the area of our square, nine over two, is equal to the side squared. If we take the square root of both sides, the square root of a side squared equals 𝑠, a side, and the side of this square measures the square root of nine over two centimeters. We can write that as the square root of nine over the square root of two. We’re only interested in the positive square root since we’re dealing with distance. So we can say the square root of nine equals three.
And we’ve now found the side length of every side of this square. But that’s in fact not what this question is looking for. We’re interested in the length of the diagonal 𝐵𝐷. Because we know that this is a square, we could say that angle 𝐵𝐶𝐷 measures 90 degrees, which means line 𝐵𝐷 is the hypotenuse of a right triangle. And we can use the Pythagorean theorem to find the length of 𝐵𝐷. Since 𝐵𝐷 is the hypotenuse, that’s the variable 𝐶 and the Pythagorean theorem, 𝐵𝐷 squared will be equal to the sum of the other two sides squared, which is three over the square root of two squared plus three over the square root of two squared.
We know that three over the square root of two squared is nine over two because we were given that to start with. And so we have nine over two plus nine over two. 𝐵𝐷 squared equals nine over two plus nine over two. Nine over two plus nine over two is 18 over two, which reduces to nine. We’ll take the square root of both sides. The square root of 𝐵𝐷 squared is just line 𝐵𝐷. And the square root of nine is plus or minus three. But since we’re dealing with distance, we only want the positive square root, which is three. And that means the diagonal 𝐵𝐶 measures three centimeters.
There is one other way we could work this out without using the Pythagorean theorem. To do this, we need to remember a few things. Number one, a square is a type of rhombus. Number two, we find the area of a rhombus by multiplying 𝑝 times 𝑞 and dividing by two where 𝑝 and 𝑞 are the diagonals of a rhombus. Number three, the diagonals of a square are equal to one another. Therefore, another way to find the area of a square is to say the diagonal squared divided by two.
And if you know all of that, you can say the area of this square is equal to 𝐵𝐷 squared over two. And then you can plug in nine over two for the area because that was given to us. If nine over two is equal to 𝐵 squared over two, then nine equals 𝐵𝐷 squared. And we take the square root of both sides of this equation to show that 𝐵𝐷 equals three. This second method requires far less calculation but depends on you remembering these facts about a square and a rhombus. But both methods confirm that the length of 𝐵𝐷, the diagonal of this square, is three centimeters.