# ISEE Upper Level Quantitative : Trapezoids

## Example Questions

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

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

13 to 6

33 to 19

20 to 13

10 to 3

33 to 19

Explanation:

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 #1 : How To Find The Area Of A Trapezoid

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

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

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

(b) is the greater quantity

Explanation:

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 #2 : How To Find The Area Of A Trapezoid

Figure NOT drawn to scale.

The above figure depicts Trapezoid  with midsegment , and .

Give the area of Trapezoid

Explanation:

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 #1 : How To Find The Length Of The Side Of A Trapezoid

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

(a)

(b) 24

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

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

(b) is the greater quantity

Explanation:

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 #51 : Quadrilaterals

Figure NOT drawn to scale.

The above figure depicts Trapezoid  with midsegment , and .

Give the area of Trapezoid  in terms of .

Explanation:

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 #1 : How To Find The Length Of The Side Of A Trapezoid

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

Explanation:

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 #11 : Trapezoids

Given Trapezoid , where  . Also,

Which is the greater quantity?

(a)

(b)

(b) is greater

(a) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

Explanation:

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 #51 : Quadrilaterals

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

Which is the greater quantity?

(a)

(b)

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

(b) is greater.

Explanation:

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 #251 : Plane Geometry

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

(a)

(b)

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

(a) and (b) are equal.

Explanation:

;  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.

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