How to find the length of the diagonal of a rectangle
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GRE Quantitative Reasoning › How to find the length of the diagonal of a rectangle
If rectangle 
Explanation
Lets call our longer side L and our shorter side W.
If the perimeter is equal to 68, then
We also have that
If we then plug this into our equation for perimeter, we get 
Therefore, 
Given Rectangle ABCD.
Quantity A: The length of diagonal AC times the length of diagonal BD
Quantity B: The square of half of ABCD's perimeter
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
Explanation
Suppose ABCD has sides a and b.
The length of one of ABCD's diagonals is given by a2+ b2 = c2, where c is one of the diagonals.
Note that both diagonals are of the same length.
Quantity A: The length of diagonal AC times the length of diagonal BD
This is c * c = c2.
Quantity A = c2 = a2+ b2
Now for Quantity B, remember that the perimeter of a rectangle with sides a and b is Perimeter = 2(a + b).
Half of Perimeter = (a + b)
Square Half of Perimeter = (a + b)2
Use FOIL: (a + b)2 = a2+ 2ab + b2
Quantity B = (a + b)2 = a2+ 2ab + b2
The question is asking us to compare a2+ b2 with a2+ 2ab + b2.
Note that as long as a and b are positive numbers (in this case a and b are dimensions of a rectangle, so they must be positive), the second quantity will be greater.