### All ISEE Upper Level Quantitative Resources

## 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

**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

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

**Possible Answers:**

(A) is greater

(A) and (B) are equal

(B) is greater

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

**Correct answer:**

(A) and (B) are equal

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 .

**Possible Answers:**

**Correct answer:**

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

.