Dimensions of a Square What is the length of ...

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Dimensions of a Square What is the length of the side of a square if its area and perimeter are numerically equal?

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Hello, today, we're going to be solving the following problem. So we want to find the length of the side of the square if its perimeter is equal to the square of its diagonal. So let's go ahead and quickly draw a picture. So we know that we're working with the square and we're going to be using the diagonal of the square. We want to find the side lengths if its perimeter is equal to the square of its diagonal. So the thing about a square is that all the sides of the square are of the same length and we don't know the side of the square. So we're going to represent one side as X and if one side is represented as X, then all the other sides are going to be represented by X as well. And we know that we're going to be working with the perimeter. So let's go ahead and find the perimeter of the square. Now, the perimeter is just the sum of all the sides combined. And that's going to give us X plus X plus X plus X and that is going to simplify to four X. So this is going to be the perimeter of the box. Furthermore, we want to work with the diagonal of the box as well. Now, the diagonal is going to be this line stretching from the upper left corner to the lower right corner. But how do we find the length of this diagonal? Well, the link we can use the diagonal and two sides of the box to represent a right triangle. And using that, we can go ahead and use the hype, the Pythagorean theorem to find the length of the diagonal. So the Pythagorean theorem states that the hypotenuse Of a triangle is equal to the sum of the two sides squared. But since every side is of the same value X, we're going to represent A and B as X. So we get C squared is equal to X squared plus X squared. And that is going to simplify to give A C squared is equal to two X squared. Now, in order to solve for C, we're going to take the square root of both sides of this equation. And that is going to leave us with C is equal to the square root of two X squared. Now the square root allows us to take out the X squared as a single X in front of the square root. So C is going to equal to X times the square root of two. So that is going to be the length of the diagonal. Now, we are told that the perimeter is equal to the square of the diagonal. And the way that we can represent that is we can let P equal to C squared. So we already know what P is. It's for X. And we know that C is equal to X times the square root of two. So we can go ahead and replace the following values into the current equation. And that is going to give us four X is equal to X times the square root of two squared. Now, on the right hand side of this equation, we're gonna go ahead and distribute the exponent of two to both X and the square root of two. But the square root of two squared is just going to cancel out the square root. So on the right hand side, what we have is four X is equal to X squared times two and that is going to equal 22 X squared. Now, since we want to solve for X, we want to set this equation equal to zero. And we can do that by subtracting four X to the right hand side of this equation that's going to give us two X squared minus four X is equal to zero. Now, in order to simplify, we can go ahead and factor out a two X from the right hand side of this equation. And what we get is two X times X minus two. And now using the zero property we can set both of these factors equal to zero in order to solve for X. So doing that, we get to X is equal to zero and we get X minus two is equal to zero. Now, on the left hand side, in order to solve for X, we're just going to divide both sides of this equation by two. And that's going to give us X is equal to zero. But the problem with this is that a side length cannot be equal to zero. So we can allow X to equal to zero. Otherwise we don't have anything to work with. So we're gonna go ahead and simplify the right hands equation. And in order to solve for X on the right hand side, we're going to add two to both sides of the equation and that is going to give us X is equal to two, which is going to be the length of each side of the square. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Dimensions of a Square What is the length of the side of a square if its area and perimeter are numerically equal?

Key Concepts

Area of a Square

Perimeter of a Square

Setting up and Solving Equations

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