ISEE Upper Level Quantitative : Trapezoids

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

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Example Questions

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Example Question #1 : Trapezoids

Trapezoid

Which quantity is greater?

(a) The perimeter of the above trapezoid

(b) The perimeter of a rectangle with length and width  and , respectively.

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:

(a) is the greater quantity

Explanation:

The perimeter of a rectangle is twice the sum of its length and its width:

Since the height of the trapezoid in the figure is , both of its legs must have length greater than or equal to . But for a leg to be of length, it must be perpendicular to the bases. Since perpendicularity of both legs would make the trapezoid a rectangle - which it cannot be - it follows that both legs cannot be of length . Therefore, the perimeter of the trapezoid is:

The perimeter of the trapezoid must be greater than that of the rectangle.

Example Question #2 : Trapezoids

Trapezoid

Figure NOT drawn to scale.

In the above figure,  is the midsegment of isosceles Trapezoid . Also, .

What is the perimeter of Trapezoid  ?

Possible Answers:

Correct answer:

Explanation:

The length of the midsegment of a trapezoid is half sum of the lengths of the bases, so

.

Also, by definition, since Trapezoid  is isosceles, . The midsegment divides both legs of Trapezoid  into congruent segments; combining these facts:

.

, so the perimeter of Trapezoid  is 

.

Example Question #3 : Trapezoids

Trapezoid

In the above figure,  is the midsegment of Trapezoid 

Which is the greater quantity?

(a) Twice the perimeter of Trapezoid 

(b) The perimeter of Trapezoid 

Possible Answers:

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

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

The midsegment of a trapezoid bisects both of its legs, so 

  and  .

For reasons that will be apparent later, we will set

Also, the length of the midsegment is half sum of the lengths of the bases:

.

The perimeter of  Trapezoid  is 

Twice this is 

The  perimeter of  Trapezoid  is

 and , so , making (a) the greater quantity.

Example Question #4 : 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:

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

Correct answer:

(a) and (b) are equal.

Explanation:

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 #5 : 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) is greater

(a) and (b) are equal

(b) is greater

It it impossible to tell from the information given

Correct answer:

(a) is greater

Explanation:

 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:

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

Correct answer:

(a) and (b) are equal.

Explanation:

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 #7 : Trapezoids

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:

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

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 #1 : Trapezoids

Trapezoid

Which quantity is greater?

(a) The area of the above trapezoid

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

Possible Answers:

(b) is the greater quantity

(a) and (b) are equal

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

(a) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

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 #9 : Trapezoids

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

(a) and (b) are equal

(b) is the greater quantity

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

Correct answer:

(a) and (b) are equal

Explanation:

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

Trapezoid

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

Possible Answers:

Correct answer:

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  - the shaded trapezoid - is

The area of Trapezoid  is

The percent of Trapezoid  that is shaded in is

 

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