The perimeter of a rectangle can be found by the following formula:   Let's substitute our values from the problem statement:

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We also know from the second line that  Let's substitute that back into the perimeter equation, and solve for width. Then, we plug that back into the other eqution to get the answer for length. 

If the rectangle has a length of  and a width of , we can imagine the diagonal as being the hypotenuse of a right triangle. The length and width are the other two sides to this triangle, so we can use the Pythagorean Theorem to calculate the length of the diagonal of the rectangle:

The diagonal of a rectangle can be thought of as the hypotenuse of a right triangle with the length and width of the rectangle as the other two sides. This means we can use the Pythagorean theorem to solve for the diagonal of a rectangle if we are given its length and its width:

Here, we also have a  triangle, indeed both ADC and ABD are   triangles. We can see that by calculating the missing angles. in each triangle we find that this angle to be 30 degrees. Since , we also know that the other sides of the triangles will be  and , since a   triangle have its sides in ratio , where  is a constant. In this case . Therefore the hypotenuse will be , which is also the length of the diagonal of the rectangle.

Firstly, before we try to set up an equation for the length of the sides and the area, we should notice that the area is a perfect square. Indeed . Now let's try to see whether 13 could be the length of two consecutive sides of the rectangle, Indeed, we are told that , therefore ABDC is a square with side 13 and with diagonal , where  is the length of the side of the square. Therefore, the final answer is .

To find the length of the sides to calculate the length of the diagonal with the Pythagorean Theorem, we would need to set up two equations for our variables. However, trial and error, in my opinion in most GMAT problems is faster than trying to solve a quadratic equation. The way we should test values for our sides is firstly by finding the different possible factors of the area, that way we can see possible factors.

As follows  has possible factors . From these values we should find the two that will give us 8, the length of the two consecutive sides.

We find that  and  are the values for the two sides.

Now we just need to apply the Pythagorean Theorem to find the length of the diagonal: ,  or .

To find the diagonal, you must use the pythaorean theorem. Thus:

To solve, simply use the Pythagorean theorem.

If a rectangular garden bed will have side lengths of 9 meters and 12 meters, what will the distance be across its diagonal?

This question is a rectangle question, but it could also be seen as a triangle question. If we have a rectangle with two side lengths, we can find the diagonal by using Pythagorean theorem:

If you are feeling really observant, you may have seen that we have a Pythagorean triple. In this case, a 3-4-5 triangle. You could have skipped using Pythagoran theorem and simply done three times five to get fifteen meters.

If the area of the rectangle is equal to the area of the square, then it must have an area of  . If the rectangle has an area of  and a side with a lenth of 3 inches, then the equation to solve the problem would be , where  is the length of the rectangle. The solution:

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