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ISEE Upper Level Quantitative : Geometry

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All ISEE Upper Level Quantitative Resources

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Example Questions

Example Question #1 : Pentagons

A pentagon with a perimeter of one mile has three congruent sides. The second-longest side is 250 feet longer than any one of those three congruent sides, and the longest side is 500 feet longer than the second-longest side.

Which is the greater quantity?

(a) The length of the longest side of the pentagon

(b) Twice the length of one of the three shortest sides of the pentagon

Possible Answers:

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

(a) is greater.

Correct answer:

(b) is greater.

Explanation:

If each of the five congruent sides has measure , then the other two sides have measures x + 250 and $\left (x + 250 \right ) +500 = x + 750$ . Add the sides to get the perimeter, which is equal to 5,280 feet, the solve for :

x + x + x + (x+250) + (x+ 750) = 5,280

5x + 1,000 = 5,280

5x + 1,000 - 1,000 = 5,280- 1,000

5x = 4,280

$5x\div 5 = 4,280 \div 5$

x = 856 feet

Now we can compare (a) and (b).

(a) The longest side has measure x + 750= 856 + 750 = 1,606 feet.

(b) The three shortest sides each have length 856 feet; twice this is $856 \times 2 = 1,712$ feet.

(b) is greater.

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Example Question #1 : How To Find The Length Of The Side Of A Pentagon

A regular pentagon has perimeter one yard. Which is the greater quantity?

(A) The length of one side

(B) 7 inches

Possible Answers:

(A) and (B) are equal

(A) is greater

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

(B) is greater

Correct answer:

(A) is greater

Explanation:

One yard is equal to 36 inches. A regular pentagon has five sides of equal length, so one side of the pentagon has length

$36 \div 5 = 7 \frac{1}{5}$ inches.

Since $7 \frac{1}{5} > 7$ , (A) is greater.

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Example Question #1 : Pentagons

The sum of the lengths of three sides of a regular pentagon is one foot. Give the perimeter of the pentagon in inches.

Possible Answers:

It is impossible to determine the perimeter from the information given.

$15\; \textrm{in}$

$25\; \textrm{in}$

$20 \; \textrm{in}$

$30 \; \textrm{in}$

Correct answer:

$20 \; \textrm{in}$

Explanation:

A regular pentagon has five sides of the same length.

One foot is equal to twelve inches; since the sum of the lengths of three of the congruent sides is twelve inches, each side measures

$12 \div 3 = 4$ inches.

The perimeter is

$4 \times 5 = 20$ inches.

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Example Question #2 : Pentagons

One side of a regular pentagon is 20% longer than one side of a regular hexagon. Which is the greater quantity?

(A) The perimeter of the pentagon

(B) The perimeter of the hexagon

Possible Answers:

(B) is greater

(A) is greater

(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:

Let be the length of one side of the hexagon. Then its perimeter is .

Each side of the pentagon is 20% greater than this length, or

s + 0.20s = 1.20 s .

The perimeter is five times this, or $5 \cdot 1.2s = 6s$ .

The perimeters are the same.

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Example Question #3 : Pentagons

The length of one side of a regular octagon is 60% of that of one side of a regular pentagon. What percent of the perimeter of the pentagon is the perimeter of the octagon?

Possible Answers:

$92 \%$

It is impossible to answer the question from the information given.

$96 \%$

$120 \%$

$111 \%$

Correct answer:

$96 \%$

Explanation:

Let be the length of one side of the regular pentagon. Then its perimeter is .

The length of one side of the regular octagon is 60% of , or 0.6s , so its perimeter is $8 \cdot 0.6s = 4.8 s$ .The answer is therefore the percent 4.8s is of , which is

$\frac{4.8s}{5s} \times 100 = 0.96 \times 100 = 96 \%$

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Example Question #3 : Pentagons

One side of a regular hexagon is 20% shorter than one side of a regular pentagon. Which is the greater quantity?

(A) The perimeter of the pentagon

(B) The perimeter of the hexagon

Possible Answers:

(A) is greater

(A) and (B) are equal

(B) is greater

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

Correct answer:

(A) is greater

Explanation:

Let be the length of one side of the pentagon. Then its perimeter is .

Each side of the hexagon is 20% less than this length, or

s - 0.20s = 0.80 s .

The perimeter is five times this, or $6 \cdot 0.8s = 4.8s$ .

Since 5 > 4.8 and is positive, 5s > 4.8s , so the pentagon has greater perimeter, and (A) is greater.

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Example Question #2 : Pentagons

A pentagon has five angles whose measures are $x^{\circ}, x^{\circ}, x^{\circ}, y^{\circ}, 2y^{\circ}$ .

Which quantity is greater?

(a) x + y

(b) 180

Possible Answers:

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

Correct answer:

(a) and (b) are equal

Explanation:

The angles of a pentagon measure a total of 180 (5-2) = 540 . From the information, we know that:

x + x + x + y + 2y = 540

3x + 3y = 540

$3\left ( x + y \right )= 540$

$3\left ( x + y \right )\div 3= 540\div 3$

x + y = 180

making the two quantities equal.

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Example Question #4 : Pentagons

A pentagon has five angles whose measures are $x^{\circ}, x^{\circ}, x^{\circ}, y^{\circ}, y^{\circ}$ .

Which quantity is greater?

(a)

(b)

Possible Answers:

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

Correct answer:

It is impossible to tell from the information given.

Explanation:

The angles of a pentagon measure a total of 180 (5-2) = 540 . From the information given, we know that:

x + x + x + y + y = 540

3x + 2y = 540

However, we cannot tell whether or is greater. For example, if x = 90 , then y = 135 ; if x = 140 , then y = 60 .

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Example Question #3 : Pentagons

Pentagon ABCDE and hexagon UVWXYZ are both regular and have equal sidelengths. Diagonals $\overline{AC}$ and $\overline{UW }$ are constructed.

Which is the greater quantity?

(a) $m \angle BAC$

(b) $m \angle VUW$

Possible Answers:

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

In both situations, the two adjacent sides and the diagonal form an isosceles triangle.

By the Isosceles Triangle Theorem, $m \angle BAC = m \angle BCA$ and $m \angle VUW = m \angle VWU$ . Also, since the measures of the angles of a triangle total $180^{\circ }$ , we know that

$m \angle BAC + m \angle BCA + m \angle ABC = 180$

and

$m \angle VWU + m \angle VUW + m \angle UVW = 180$ .

We can use these equations to compare $m \angle BAC$ and $m \angle VUW$ .

(a) $m \angle ABC = \frac{ 180 (5-2)}{5} = 108 ^{\circ }$

$m \angle BAC + m \angle BCA + m \angle ABC = 180 ^{\circ }$

$m \angle BAC + m \angle BAC + 108 ^{\circ } = 180 ^{\circ }$

$2\cdot m \angle BAC + 108 ^{\circ }= 180 ^{\circ }$

$2\cdot m \angle BAC + 108 ^{\circ } -108 ^{\circ } = 180 ^{\circ }-108 ^{\circ }$

$2\cdot m \angle BAC = 72 ^{\circ }$

$2 \cdot m \angle BAC \div 2 = 72 ^{\circ } \div 2$

$m \angle BAC = 36 ^{\circ }$

(b) $m \angle UVW = \frac{ 180 (6-2)}{6} = 120 ^{\circ }$

$m \angle VWU + m \angle VUW + m \angle UVW = 180 ^{\circ }$

$m \angle VUW + m \angle VUW + 120^{\circ } = 180 ^{\circ }$

$2 \cdot m \angle VUW + 120^{\circ } = 180 ^{\circ }$

$2 \cdot m \angle VUW + 120^{\circ } -120^{\circ } = 180 ^{\circ }-120^{\circ }$

$2 \cdot m \angle VUW = 60^{\circ }$

$2 \cdot m \angle VUW \div 2 = 60^{\circ }\div 2$

$m \angle VUW = 30^{\circ }$

$m \angle BAC > m \angle VUW$

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Example Question #9 : Pentagons

Pentagon ABCDE and hexagon UVWXYZ are both regular, with their sidelengths equal. Diagonals $\overline{AC}$ and $\overline{UW }$ are constructed.

Which is the greater quantity?

(a)

(b)

Possible Answers:

(b) is greater.

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Correct answer:

(b) is greater.

Explanation:

Each diagonal, along with two consecutive sides of its polygon, forms a triangle. All of the sides of the pentagon and the hexagon are congruent to one another, so between the two triangles, there are two pairs of two congruent corresponding sides:

$\overline{AB} \cong \overline{ UV }$

$\overline{BC} \cong \overline{ VW }$

Their included angles, $\angle B$ and $\angle V$ , are interior angles of the pentagon and hexagon, respectively. The angle with greater measure will be opposite the longer side. We can use the Interior Angles Theorem to calculate the measures:

$m\angle B =\frac{ 180 (5-2)}{5} =108 ^{\circ }$

$m\angle V =\frac{ 180 (6-2)}{6} =120 ^{\circ }$

$m\angle V > m\angle B \Rightarrow UW > AC$

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Geometry

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : Pentagons

Example Question #1 : How To Find The Length Of The Side Of A Pentagon

Example Question #1 : Pentagons

Example Question #2 : Pentagons

Example Question #3 : Pentagons

Example Question #3 : Pentagons

Example Question #2 : Pentagons

Example Question #4 : Pentagons

Example Question #3 : Pentagons

Example Question #9 : Pentagons

All ISEE Upper Level Quantitative Resources

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