### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #11 : Trapezoids

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

**Possible Answers:**

13 to 6

10 to 3

20 to 13

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 #12 : Trapezoids

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:**

(b) is the greater quantity

(a) 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

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 #13 : Trapezoids

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.

### Example Question #14 : Trapezoids

In the above diagram, which depicts Trapezoid , and . Which is the greater quantity?

(a)

(b) 24

**Possible Answers:**

**Correct answer:**

(b) is the greater quantity

To see that (b) is the greater quantity of the two, it suffices to construct the midsegment of the trapezoid - the segment which has as its endpoints the midpoints of legs and . Since and , the midsegment, , is positioned as follows:

The length of the midsegment is half the sum of the bases, so

, so .

### Example Question #15 : Trapezoids

Figure NOT drawn to scale.

The above figure depicts Trapezoid with midsegment . , and .

Give the area of Trapezoid in terms of .

**Possible Answers:**

**Correct answer:**

The midsegment of a trapezoid has as its length half the sum of the lengths of the bases, which here are and :

Therefore,

The area of Trapezoid is one half multiplied by the height, , multiplied by the sum of the lengths of the bases, and . The midsegment of a trapezoid bisects both legs, so , and the area is

### Example Question #16 : Trapezoids

The above figure depicts Trapezoid with midsegment . Express in terms of .

**Possible Answers:**

**Correct answer:**

The midsegment of a trapezoid has as its length half the sum of the lengths of the bases, which here are and :

The correct choice is .

### Example Question #17 : Trapezoids

Given Trapezoid , where . Also,

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

(b) is greater

**Correct answer:**

(a) is greater

and are same-side interior angles, as are and .

The Same-Side Interior Angles Theorem states that if two parallel lines are crossed by a transversal, then the sum of the measures of a pair of same-side interior angles is always . Therefore,

, or

, or

Substitute:

(a) is the greater quantity

### Example Question #18 : Trapezoids

Consider trapezoid , where . Also, is acute and is obtuse.

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

**Correct answer:**

(b) is greater.

and are same-side interior angles, as are and .

The Same-Side Interior Angles Theorem states that if two parallel lines are crossed by a transversal, then same-side interior angles are supplementary. A pair of supplementary angles comprises either two right angles, or one acute angle and one obtuse angle. Since is acute and is obtuse, is obtuse and is acute. Therefore the greater measure of the two, making (b) greater.

### Example Question #19 : Trapezoids

The above diagram depicts trapezoid . Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

**Correct answer:**

(a) and (b) are equal.

; and are same-side interior angles, as are and .

The Same-Side Interior Angles Theorem states that if two parallel lines are crossed by a transversal, then the sum of the measures of a pair of same-side interior angles is always .

Therefore, , making the two quantities equal.