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## Example Questions

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

Note: Figure NOT drawn to scale.

Refer to the above figure, which shows a rectangle divided into two smaller rectangles.

True or false:.

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

The statements are actually equivalent. Since , Statement 2 can be rewritten as , or - and, since all quantities are positive, . The question is therefore whether either statement alone answers the question or both together do not.

Each statement is equivalent to

.

Divide both sides by to yield a proportion statement:

The sides of the rectangles are in proportion; subsequently, the rectangles are similar.

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

You are given two rectangles, and

True or false: .

Statement 1:

Statement 2:

**Possible Answers:**

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Assume Statement 1 alone. Divide both sides by ,

.

This proportion statement asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.

Now examine Statement 2.

By the congruence of opposite sides of a rectangle,

, ,

and, regardless of whether the rectangles are similar or not,

.

Therefore, Statement 2 provides superfluous and unhelpful information.

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

You are given two rectangles, and .

Let the perimeter of be , and let the perimeter of be .

True or false: .

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

The perimeter of is the sum of its sidelengths, and opposite sides are congruent, so

Similarly,

Therefore,

,

and, reducing,

Assume Statement 1 alone. Then

, or , and by a property of proportions,

Therefore,

,

thereby proving the sides of the rectangles to be in proportion. As a consequence, .

By a similar argument, Statement 2 also proves .

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

You are given two rectangles, and .

Let the perimeter of be , and let the perimeter of be .

Let the area of be and the area be .

True or false: .

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Assume Statement 1 alone.

Examine these three rectangles. The one of the left is ; the other two have the same dimensions, and both are called , except that the names of the vertices are differently arranged:

Regardless of which is chosen, the ratio of the perimeters is and the ratio of the areas is . The conditions of the problem are met for both pairings, but in one case, and in the other, (that is, the rectangles are similar, but the given similarity statement may or may not be true).

Assume Statement 2 alone.

The perimeter of is the sum of its sidelengths, and opposite sides are congruent, so

Similarly,

Therefore,

,

and, reducing,

From Statement 2,

, or , and by a property of proportions,

Therefore,

,

thereby proving the sides of the rectangles to be in proportion. As a consequence, .

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

You are given two rectangles, and

True or false: .

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

Statement 1 compares the lengths of the diagonals of the two rectangles. Since the diagonals of any rectangle are congruent, , , and, as a consequence, regardless of whether the rectangles are similar or not. Statement 1 is a superfluous statement and is therefore unhelpful.

Statement 2 asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.

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

You are given two rectangles, and .

Let the perimeter of be , and let the perimeter of be .

Let the area of be and the area be .

True or false: .

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

Assume both statements to be true.

Examine these three rectangles. The one of the left is ; the other two have the same dimensions, and both are called , except that the names of the vertices are differently arranged:

Regardless of which is chosen:

The ratio of the perimeters is ;

The ratio of the areas is ; and,

The ratio .

The conditions of the problem are met for both pairings, but in one case, and in the other, (that is, the rectangles are similar but the similarity statement *given* may be true or false).

The two statements together provide insufficient information.

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