## Example Questions

### Example Question #1 : Rectangles

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.

The relationship cannot be determined from the information given.

Quantity B is greater.

The two quantities are equal.

Quantity B is greater.

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.

### Example Question #5 : Rectangles

If rectangle has a perimeter of , and the longer edge is times longer than the shorter edge, then how long is the diagonal ?      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, and . Using the Pythagorean Theorem, we have so .

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