All ISEE Upper Level Quantitative Resources
Example Questions
Example Question #1 : How To Find The Area Of A Trapezoid
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
(a) is greater.
It is impossible to tell from the information given.
(a) and (b) are equal.
(b) is greater.
(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
It it impossible to tell from the information given
(b) is greater
(a) is greater
(a) and (b) are equal
(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 #3 : How To Find The Area Of A Trapezoid
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.
(a) and (b) are equal.
(b) is greater.
It is impossible to tell from the information given.
(a) is greater.
(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 #2 : 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.
It is impossible to tell from the information given.
(a) is greater.
(b) is greater.
(a) and (b) are equal.
(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 : 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 sides of length
(b) is the greater quantity
It is impossible to determine which is greater from the information given
(a) is the greater quantity
(a) and (b) are equal
(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 #6 : 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
(a) is the greater quantity
(b) is the greater quantity
(a) and (b) are equal
It is impossible to determine which is greater from the information given
(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 #7 : How To Find The Area Of A Trapezoid
In the above figure, is the midsegment of Trapezoid . What percent of Trapezoid has been shaded in?
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 #8 : 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
10 to 3
20 to 13
33 to 19
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 #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
(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
(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 #9 : 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 .
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.
Certified Tutor
Certified Tutor