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## Example Questions

### Example Question #1 : How To Find If Rectangles Are Similar

The perimeter of a rectangle is 14, and the diagonal connecting two vertices is 5.

Quantity A: 13

Quantity B: The area of the rectangle

**Possible Answers:**

The two quantities are equal.

Quantity B is greater.

The relationship between A and B cannot be determined.

Quantity A is greater.

**Correct answer:**

Quantity A is greater.

One potentially helpful first step is to draw the rectangle described in the problem statement:

After that, it's a matter of using the other information given. The perimeter is given as 14, and can be written in terms of the length and width of the rectangle:

Furthermore, notice that the diagonal forms the hypotenuse of a right triangle. The Pythagorean Theorem may be applied:

This provides two equations and two unknowns. Redefining the first equation to isolate gives:

Plugging this into the second equation in turn gives:

Which can be reduced to:

or

Note that there are two possibile values for ; 3 or 4. The one chosen is irrelevant. Choosing a value 3, it is possible to then find a value for :

This in turn allows for the definition of the rectangle's area:

So Quantity B is 12, which is less than Quantity A.

### Example Question #14 : Rectangles

One rectangle has sides of and . Which of the following pairs could be the sides of a rectangle similar to this one?

**Possible Answers:**

and

and

and

and

**Correct answer:**

and

For this problem, you need to find the pair of sides that would reduce to the same ratio as the original set of sides. This is a little tricky at first, but consider the set:

and

For this, you have:

Now, if you factor out , you have:

Thus, the proportions are the same, meaning that the two rectangles would be similar.

### Example Question #2 : How To Find If Rectangles Are Similar

One rectangle has a height of and a width of . Which of the following is a possible perimeter of a similar rectangle, having one side that is ?

**Possible Answers:**

**Correct answer:**

Based on the information given, we know that could be *either* the longer *or* the shorter side of the similar rectangle. Similar rectangles have proportional sides. We might need to test both, but let us begin with the easier proportion, namely:

as

For this proportion, you really do not even need fractions. You know that must be .

This means that the figure would have a perimeter of

Luckily, this is one of the answers!

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