# ISEE Upper Level Quantitative : Parallelograms

## Example Questions

### Example Question #1 : How To Find The Area Of A Parallelogram

In the above parallelogram,  is acute. Which is the greater quantity?

(A) The area of the parallelogram

(B) 120 square inches

(A) is greater

(B) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

(B) is greater

Explanation:

Since  is acute, a right triangle can be constructed with an altitude as one leg and a side as the hypotenuse, as is shown here. The height of the triangle must be less than its sidelength of 8 inches.

The height of the parallelogram must be less than its sidelength of 8 inches.

The area of the parallelogram is the product of the base and the height - which is

Therefore,

(B) is greater.

### Example Question #2 : Parallelograms

Parallelogram A is below:

Parallelogram B is below:

Note: These figures are NOT drawn to scale.

Refer to the parallelograms above. Which is the greater quantity?

(A) The area of parallelogram A

(B) The area of parallelogram B

(A) is greater

(A) and (B) are equal

(B) is greater

It is impossible to determine which is greater from the information given

(A) and (B) are equal

Explanation:

The area of a parallelogram is the product of its height and its base; its slant length is irrelevant. Both parallelograms have the same height (8 inches) and the same base (1 foot, or 12 inches), so they have the same areas.

### Example Question #3 : Parallelograms

Figure NOT drawn to scale

The above figure shows Rhombus  and  are midpoints of their respective sides. Rectangle  has area 150.

Give the area of Rhombus .

Explanation:

A rhombus, by definition, has four sides of equal length. Therefore, . Also, since  and  are the midpoints of their respective sides,

We will assign  to the common length of the four half-sides of the rhombus.

Also, both  and  are altitudes of the rhombus; the are congruent, and we will call their common length  (height).

The figure, with the lengths, is below.

Rectangle  has dimensions  and ; its area, 150, is the product of these dimensions, so

The area of the entire Rhombus  is the product of its height  and the length of a base , so

.

### Example Question #4 : Parallelograms

In the above parallelogram,  is acute. Which is the greater quantity?

(A) The perimeter of the parallelogram

(B) 46 inches

(A) and (B) are equal

(B) is greater

(A) is greater

It is impossible to determine which is greater from the information given

(A) and (B) are equal

Explanation:

The measure of  is actually irrelevant. The perimeter of the parallelogram is the sum of its four sides; since opposite sides of a parallelogram have the same length, the perimeter is

inches,

making the quantities equal.

### Example Question #61 : Quadrilaterals

Parallelogram A is below:

Parallelogram B is below:

Note: These figures are NOT drawn to scale.

Refer to the parallelograms above. Which is the greater quantity?

(A) The perimeter of parallelogram A

(B) The perimeter of parallelogram B

(B) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

(A) is greater

(A) is greater

Explanation:

The perimeter of a parallelogram is the sum of its sidelengths; its height is irrelevant. Also, opposite sides of a parallelogram are congruent.

The perimeter of parallelogram A is

inches;

The perimeter of parallelogram B is

inches.

(A) is greater.

### Example Question #6 : Parallelograms

Figure NOT drawn to scale.

The above figure depicts Rhombus  with  and .

Give the perimeter of Rhombus .

Explanation:

All four sides of a rhombus have the same length, so we can find the perimeter of Rhombus  by taking the length of one side and multiplying it by four. Since , the perimeter is four times this, or .

Note that the length of  is actually irrelevant to the problem.

### Example Question #7 : Parallelograms

In Parallelogram , and . Which of the following is greater?

(A)

(B)

(a) and (b) are equal

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(b) is the greater quantity

It cannot be determined which of (a) and (b) is greater

Explanation:

In Parallelogram , and  are adjoining sides; there is no specific rule for the relationship between their lengths. Therefore, no conclusion can be drawn of and , and no conclusion can be drawn of the relationship between and .

### Example Question #8 : Parallelograms

Which of the following can be the measures of the four angles of a parallelogram?

Explanation:

Opposite angles of a parallelogram must have the same measure, so the correct choice must have two pairs, each of the same angle measure. We can therefore eliminate  and  as choices.

Also, the sum of the measures of the angles of any quadrilateral must be , so we add the angle measures of the remaining choices:

, so we can eliminate this choice.

:

, so we can eliminate this choice.

; this is the correct choice.

### Example Question #9 : Parallelograms

Refer to the above figure, which shows a parallelogram. What is  equal to?

Not enough information is given to answer this question.

Explanation:

The sum of two consecutive angles of a parallelogram is .

157 is the correct choice.

### Example Question #10 : Parallelograms

In Parallelogram , and .

Which is the greater quantity?

(a)

(b)

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(b) is the greater quantity

(a) and (b) are equal