# GMAT Math : Calculating whether rectangles are similar

## Example Questions

### Example Question #1 : Calculating Whether Rectangles Are Similar

Note: Figure NOT drawn to scale.

Refer to the above figure. .

.

Give the ratio of the area of the shaded region to the area of .

Explanation:

The ratio of the areas of similar rectangles is the square of the similarity ratio, so the ratio of the area of  to that of  is

.

So if the area of  is , the area of  is

The area of the shaded region is the difference between the areas of the rectangles, making this area

.

The desired ratio is 24 to 25.

### Example Question #2 : Calculating Whether Rectangles Are Similar

.

Which of the following must be equal to ?

The area of  divided by the area of

None of the other choices gives a correct answer.

The perimeter of  divided by the perimeter of

Explanation:

, so ; this is the similarity ratio of  to .

and  are not corresponding sides, nor are they congruent to corresponding sides, so it may or may not be true that .

The ratio of the perimeters of similar rectangles is the same as their actual similarity ratio, but the choice gives the rectangles in the original order; the quotient of the perimeters as given is 3, not .

The ratio of the areas of similar rectangles is the square of the similarity ratio, so the quotient in the given choice is .

However, by similarity, and the fact that opposite sides  and  are congruent,

.

The correct choice is .

### Example Question #3 : Calculating Whether Rectangles Are Similar

In the above diagram,

.

and . Give the area of .

Explanation:

, so

The area of the rectangle is

### Example Question #4 : Calculating Whether Rectangles Are Similar

, and .

All the following quantities MUST be equal to 2 except for __________.

The perimeter of  divided by the perimeter of

All of the quantities in the other choices must be equal to .

Explanation:

The two rectangles are similar with similarity ratio 2.

Corresponding sides of similar rectangles are in proportion, so

.

Since opposite sides of a rectangle are congruent, , so

.

and  are diagonals of the rectangle. If they are constructed, then, since  and  (both are right angles), by the Side-Angle-Side Similarity Theorem, . By similarity, .

The ratio of the perimeters of the rectangles is

,

It follows from  and a property of proportions that this ratio is equal to .

However,

is not a ratio of corresponding sides of the rectangle, so it does not have any restrictions on it. This is the correct choice.

### Example Question #5 : Calculating Whether Rectangles Are Similar

A certain rectangle has a length of  and a width of .  Which of the following dimensions would another rectangle need in order for it to be similar?

Explanation:

In order for two rectangles to be similar, the ratio of their dimensions must be equal. We can first check the ratio of the length to the width for the given rectangle, and then see which option has the same ratio, which will tell us whether or not the rectangle is similar:

So in order for a rectangle to be similar, the ratio of its length to its width must be the same. All we must do then is check the answer options, in no particular order, for the rectangle with the same ratio:

A rectangle with a length of  and a width of  has the same ratio of dimensions as a rectangle with a length of  and a width of , so these two rectangles are similar.

### Example Question #6 : Calculating Whether Rectangles Are Similar

An engineer is making a scale model of a building. The real building needs to have a width of  and a length of . If the engineer's scale model has a width of , what does the length of the model need to be?