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ISEE Upper Level Quantitative : Plane Geometry

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

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All ISEE Upper Level Quantitative Resources

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Example Questions

Example Question #1 : Trapezoids

Trapezoid

Figure NOT drawn to scale.

In the above figure, $\overline{XY}$ is the midsegment of isosceles Trapezoid TRAP . Also, $TX = \frac{1}{2} XY$ .

What is the perimeter of Trapezoid XYAP ?

Possible Answers:

Correct answer:

Explanation:

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

$XY = \frac{1}{2} (12 + 40) = \frac{1}{2} \cdot 52 = 26$ .

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

$TX = XP = \frac{1}{2} TP = \frac{1}{2} RA = RY = YA$

$TX = \frac{1}{2} XY = \frac{1}{2} \cdot 26 = 13$ .

XP = YA= TX = 13 , so the perimeter of Trapezoid XYAP is

XP+XY + YA + AP = 13+26+13+40 = 92 .

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Example Question #3 : Trapezoids

Trapezoid

In the above figure, $\overline{XY}$ is the midsegment of Trapezoid TRAP .

Which is the greater quantity?

(a) Twice the perimeter of Trapezoid TRYX

(b) The perimeter of Trapezoid XYAP

Possible Answers:

(a) is the greater quantity

(a) and (b) are equal

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

(b) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

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

TX = XP and RY = YA .

For reasons that will be apparent later, we will set

z = TX + RY = X P + YA

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

$XY = \frac{1}{2} (12 + 36) = \frac{1}{2} \cdot 48= 24$ .

The perimeter of Trapezoid TRYX is

TR + RY + YX + TX = TR + YX +( TX + RY) = 12+24+ z = 36 + z

Twice this is

2 (36 + z) = 72 + 2z

The perimeter of Trapezoid XYAP is

XY + YA + AP + XP = XY + AP + (XP+ YA) = 24 + 36 + z = 60 + z

72 > 60 and 2z > z , so 72 + 2z > 60+ z , making (a) the greater quantity.

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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

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) and (b) are equal.

Explanation:

Let be the common height of the figures.

(a) The area of Trapezoid A is $A = \frac{1}{2} (B+b) h = \frac{1}{2} (16+10) h = \frac{1}{2} (26) h= 13h$ .

(b) The area of Parallelogram B is

A = bh = 13h .

The figures have the same area.

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Example Question #241 : Plane Geometry

On Parallelogram ABCD , AB = 20 , locate point on $\overline{AB}$ such that AP = 13 ; locate point on $\overline{CD}$ such that DQ = 8 . Draw $\overline{PQ}$ .

Which is the greater quantity?

(a) The area of Quadrilateral APQD

(b) The area of Quadrilateral BPQC

Possible Answers:

It it impossible to tell from the information given

(b) is greater

(a) and (b) are equal

(a) is greater

Correct answer:

(a) is greater

Explanation:

$\overline{PQ}$ 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 APQD are $\overline{AP}$ and $\overline{DQ}$ . AP = 13, DQ = 8

(b) The bases of Trapezoid BPQC are $\overline{PB}$ and $\overline{QC}$ .

Opposite sides of a parallelogram are congruent, so since AB = 20 , DC= 20 also.

PB = AB -AP = 20 - 13 = 7

QC = DC - DQ = 20-8 = 12

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.

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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.

Possible Answers:

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

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 B = 85, b = 75, h = 100 into the formula for area:

$A = \frac{1}{2} (B + b) h$

$A = \frac{1}{2} \cdot (85 + 75 ) \cdot 100$

$A = \frac{1}{2} \cdot 160 \cdot 100$ '

A = 8,000 square centimeters

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

$A = bh = 80 \cdot 100 = 8,000$ square centimeters

The figures have the same area.

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Example Question #2 : How To Find The Area Of A Trapezoid

Which is the greater quantity?

(a) The area of a trapezoid with bases $3 \frac{1}{2}$ feet and $5 \frac{1}{4}$ feet and height one yard.

(b) The area of a parallelogram with base $4 \frac{1}{3}$ feet and height one yard.

Possible Answers:

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

(a) and (b) are equal.

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.

$3 \frac{1}{2} \times 12 = \frac{7}{2} \times 12 = 42$ inches

$5 \frac{1}{4} \times 12 = \frac{21}{4} \times 12 = 63$ inches

Substitute into the formula for the area of a trapezoid, setting B = 63, b = 42, h = 36 :

$A = \frac{1}{2} (B + b) h$

$A = \frac{1}{2} \cdot (63 + 42) \cdot 36$

$A = \frac{1}{2} \cdot 105 \cdot 36 = 1,890$ square inches

(b) The base of the parallelogram is

$4 \frac{1}{3} \times 12 = \frac{13}{3} \times 12 = 52$ .

Multiply this by the height:

$A = bh = 52 \cdot 36 = 1,872$ square inches

The trapezoid has greater area.

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Example Question #5 : How To Find The Area Of A Trapezoid

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

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

(a) is the greater quantity

(a) and (b) are equal

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 :

$A = \frac{1}{2} \cdot x \cdot (3x+x)$

$= \frac{1}{2} \cdot x \cdot (4x)$

$= \frac{1}{2} \cdot 4 \cdot x \cdot x$

$= 2x^{2}$

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

$A =( 2x) ^{2}= 2^{2}x^{2} = 4 x^{2}$

The square has the greater area.

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Example Question #6 : How To Find The Area Of A Trapezoid

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

(b) is the greater quantity

(a) and (b) are equal

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 :

$A = \frac{1}{2} \cdot x \cdot (3x+x)$

$= \frac{1}{2} \cdot x \cdot (4x)$

$= \frac{1}{2} \cdot 4 \cdot x \cdot x$

$= 2x^{2}$

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

$A = \frac{1}{2} \cdot 2x \cdot 2x$

$= \frac{1}{2} \cdot 2 \cdot 2 \cdot x \cdot x$

$= 2x^{2}$

The trapezoid and the square have equal area.

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Example Question #7 : How To Find The Area Of A Trapezoid

Trapezoid

In the above figure, $\overline{XY}$ is the midsegment of Trapezoid TRAP . What percent of Trapezoid TRAP has been shaded in?

Possible Answers:

$25 \%$

$37 \frac{1}{2} \%$

$33 \frac{1}{3} \%$

$36 \%$

Correct answer:

$37 \frac{1}{2} \%$

Explanation:

Midsegment $\overline{XY}$ divides Trapezoid TRAP 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:

$XY = \frac{1}{2} (12 + 36) = \frac{1}{2} \cdot 48= 24$

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 TRYX - the shaded trapezoid - is

$\frac{1}{2} \cdot h \cdot (TR+XY ) = \frac{1}{2} \cdot h \cdot (12+24) = \frac{1}{2} \cdot 36 \cdot h = 18 h$

The area of Trapezoid TRAP is

$\frac{1}{2} \cdot 2 h \cdot (TR+AP) = \frac{1}{2} \cdot 2 \cdot h \cdot (12+36) = \frac{1}{2} \cdot 2 \cdot 48 \cdot h = 48 h$

The percent of Trapezoid TRAP that is shaded in is

$\frac{18h}{48h} \cdot 100 \% = \frac{18}{48} \cdot 100 \% = 37 \frac{1}{2} \%$

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Example Question #8 : How To Find The Area Of A Trapezoid

Trapezoid

In the above figure, $\overline{XY}$ is the midsegment of Trapezoid TRAP . Give the ratio of the area of Trapezoid XYAP to that of Trapezoid TRYX .

Possible Answers:

13 to 6

10 to 3

20 to 13

33 to 19

Correct answer:

33 to 19

Explanation:

Midsegment $\overline{XY}$ divides Trapezoid TRAP 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:

$XY = \frac{1}{2} (12 + 40) = \frac{1}{2} \cdot 52 = 26$ .

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 TRYX is

$\frac{1}{2} \cdot h \cdot (TR+XY ) = \frac{1}{2} \cdot h \cdot (12+26 ) = \frac{1}{2} \cdot 38 \cdot h = 19 h$

The area of Trapezoid XYAP is

$\frac{1}{2} \cdot h \cdot (XY +AP) = \frac{1}{2} \cdot h \cdot ( 26+40 ) = \frac{1}{2} \cdot 66 \cdot h = 33 h$

The ratio of the areas is

$\frac{33 h}{19h} = \frac{33}{19}$ , or 33 to 19.

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Plane Geometry

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : Trapezoids

Example Question #3 : Trapezoids

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

Example Question #241 : Plane Geometry

Example Question #3 : How To Find The Area Of A Trapezoid

Example Question #2 : How To Find The Area Of A Trapezoid

Example Question #5 : How To Find The Area Of A Trapezoid

Example Question #6 : How To Find The Area Of A Trapezoid

Example Question #7 : How To Find The Area Of A Trapezoid

Example Question #8 : How To Find The Area Of A Trapezoid

All ISEE Upper Level Quantitative Resources

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