How to find the length of the diagonal of a rectangle - GRE Quantitative Reasoning
Card 0 of 2
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
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
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.
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.
Compare your answer with the correct one above
If rectangle 
has a perimeter of 
, and the longer edge is 
times longer than the shorter edge, then how long is the diagonal 
?
If rectangle
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, 
and 
. Using the Pythagorean Theorem, we have 
so 
.
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,
Compare your answer with the correct one above