### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #1 : Trapezoids

Trapezoid A and Parallelogram B have the same height. Trapezoid A has bases 10 and 16; Parallelogram B has base 13. Which is the greater quantity?

(a) The area of Trapezoid A

(b) The area of Parallelogram B

**Possible Answers:**

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

(b) is greater.

**Correct answer:**

(a) and (b) are equal.

Let be the common height of the figures.

(a) The area of Trapezoid A is .

(b) The area of Parallelogram B is

.

The figures have the same area.

### Example Question #2 : Trapezoids

On Parallelogram , , locate point on such that ; locate point on such that . Draw .

Which is the greater quantity?

(a) The area of Quadrilateral

(b) The area of Quadrilateral

**Possible Answers:**

(a) and (b) are equal

It it impossible to tell from the information given

(b) is greater

(a) is greater

**Correct answer:**

(a) is greater

divides the parallelogram into two trapezoids, each of which has the same height as the original parallelogram, which we will call .

(a) The bases of Trapezoid are and .

(b) The bases of Trapezoid are and .

Opposite sides of a parallelogram are congruent, so since , also.

The sum of the bases of Trapezoid A is 21; the sum of those of Trapezoid B is 19. The two trapezoids have the same height. Thereforee, since the area is one-half times the height times the sum of the bases, Trapezoid A will have the greater area.

### Example Question #1 : Trapezoids

Which is the greater quantity?

(a) The area of a trapezoid with bases 75 centimeters and 85 centimeters and height one meter.

(b) The area of a parallelogram with base 8 decimeters and height one meter.

**Possible Answers:**

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

**Correct answer:**

(a) and (b) are equal.

The easiet way to compare is to convert each measure to centimeters and calculate the areas in square centimeters. Both figures have height one meter, or 100 centimeters.

(a) Substitute into the formula for area:

'

square centimeters

(b) 8 decimeters is equal to 80 centimeters, so multiply this base by a height of 100 centimeters:

square centimeters

The figures have the same area.

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

Which is the greater quantity?

(a) The area of a trapezoid with bases feet and feet and height one yard.

(b) The area of a parallelogram with base feet and height one yard.

**Possible Answers:**

(b) is greater.

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

**Correct answer:**

(a) is greater.

The easiest way to compare the areas might be to convert each of the dimensions to inches.

(a) The bases convert by multiplying the number of feet by twelve; the height is one yard, which is 36 inches.

inches

inches

Substitute into the formula for the area of a trapezoid, setting :

square inches

(b) The base of the parallelogram is

.

Multiply this by the height:

square inches

The trapezoid has greater area.

### Example Question #5 : Trapezoids

Which quantity is greater?

(a) The area of the above trapezoid

(b) The area of a square with sides of length

**Possible Answers:**

(a) is the greater quantity

(b) is the greater quantity

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

(a) and (b) are equal

**Correct answer:**

(b) is the greater quantity

The area of a trapezoid is half the product of its height, which here is , and the sum of the lengths of its bases, which here are and :

The area of a square is the square of the length of a side, which here is :

The square has the greater area.

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

Which quantity is greater?

(a) The area of the above trapezoid

(b) The area of a square with diagonals of length

**Possible Answers:**

(a) is the greater quantity

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

(a) and (b) are equal

(b) is the greater quantity

**Correct answer:**

(a) and (b) are equal

The area of a trapezoid is half the product of its height, which here is , and the sum of the lengths of its bases, which here are and :

The area of a square, it being a rhombus, is half the product of the lengths of its diagonals, both of which are here:

The trapezoid and the square have equal area.

### Example Question #10 : Trapezoids

In the above figure, is the midsegment of Trapezoid . What percent of Trapezoid has been shaded in?

**Possible Answers:**

**Correct answer:**

Midsegment divides Trapezoid into two trapezoids of the same height, which we will call ; the length of the midsegment is half sum of the lengths of the bases:

The area of a trapezoid is one half multiplied by its height multiplied by the sum of the lengths of its bases. Therefore, the area of Trapezoid - the shaded trapezoid - is

The area of Trapezoid is

The percent of Trapezoid that is shaded in is

### Example Question #241 : Geometry

In the above figure, is the midsegment of Trapezoid . Give the ratio of the area of Trapezoid to that of Trapezoid .

**Possible Answers:**

20 to 13

13 to 6

10 to 3

33 to 19

**Correct answer:**

33 to 19

Midsegment divides Trapezoid into two trapezoids of the same height, which we will call ; the length of the midsegment is half sum of the lengths of the bases:

.

The area of a trapezoid is one half multiplied by its height multiplied by the sum of the lengths of its bases. Therefore, the area of Trapezoid is

The area of Trapezoid is

The ratio of the areas is

, or 33 to 19.

### Example Question #41 : Quadrilaterals

In the above figure, is the midsegment of Trapezoid .

Which is the greater quantity?

(a) Three times the area of Trapezoid

(b) Twice the area of Trapezoid

**Possible Answers:**

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

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

**Correct answer:**

(b) is the greater quantity

Midsegment divides Trapezoid into two trapezoids of the same height, which we will call ; the length of the midsegment is half sum of the lengths of the bases:

The area of a trapezoid is one half multiplied by its height multiplied by the sum of the lengths of its bases. Therefore, the area of Trapezoid is

.

Three times this is

.

The area of Trapezoid is, similarly,

Twice this is

.

That makes (b) the greater quantity.

### Example Question #42 : Quadrilaterals

Figure NOT drawn to scale.

The above figure depicts Trapezoid with midsegment . , and .

Give the area of Trapezoid .

**Possible Answers:**

**Correct answer:**

One way to calculate the area of a trapezoid is to multiply the length of its midsegment, which is 20, and its height, which here is

Midsegment bisects both legs of Trapezoid , in particular, . Since , .

Therefore, the area of the trapezoid is

Note that the length of is irrelevant to the problem.