ISEE Upper Level Quantitative : Parallelograms

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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Example Questions

Example Question #1 : Parallelograms

Parallelogram

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

(A) The area of the parallelogram

(B) 120 square inches

Possible Answers:

(A) is greater

(B) is greater

(A) and (B) are equal

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

Correct answer:

(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.

Parallelogram

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 #255 : Plane Geometry

Parallelogram A is below:

Rhombus_1

Parallelogram B is below:

Parallelogram

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

Possible Answers:

(B) is greater

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

(A) is greater

(A) and (B) are equal

Correct answer:

(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 #1 : How To Find The Area Of A Parallelogram

Parallelogram

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 .

Possible Answers:

Correct answer:

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.

Rhombus

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 #1 : Parallelograms

Parallelogram

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

(A) The perimeter of the parallelogram

(B) 46 inches

Possible Answers:

(A) is greater

(B) is greater

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

(A) and (B) are equal

Correct answer:

(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 #2 : Parallelograms

Parallelogram A is below:

Rhombus_1

Parallelogram B is below:

 

Parallelogram

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

Possible Answers:

(A) and (B) are equal

(B) is greater

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

(A) is greater

Correct answer:

(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 #3 : Parallelograms

Parallelogram

Figure NOT drawn to scale.

The above figure depicts Rhombus  with  and .

Give the perimeter of Rhombus .

Possible Answers:

Correct answer:

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 #4 : Parallelograms

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

(A)

(B)

Possible Answers:

(b) is the greater quantity

(a) is the greater quantity

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

(a) and (b) are equal

Correct answer:

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 #1 : Parallelograms

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

Possible Answers:

Correct answer:

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 #6 : Parallelograms

Parallelogram

 

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

Possible Answers:

Not enough information is given to answer this question.

Correct answer:

Explanation:

The sum of two consecutive angles of a parallelogram is .

157 is the correct choice.

Example Question #7 : Parallelograms

In Parallelogram , and .

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

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

Correct answer:

(b) is the greater quantity

Explanation:

In Parallelogram , and are opposite angles and are therefore congruent. This means that

Both are positive, so .

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