ISEE Upper Level Quantitative : Pentagons

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

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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:

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

Correct answer:

(b) is greater.

Explanation:

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

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

\(\displaystyle 5x + 1,000 = 5,280\)

\(\displaystyle 5x + 1,000 - 1,000 = 5,280- 1,000\)

\(\displaystyle 5x = 4,280\)

\(\displaystyle 5x\div 5 = 4,280 \div 5\)

\(\displaystyle x = 856\) feet

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

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

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

(b) is greater.

Example Question #121 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

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

(B) is greater

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

(A) 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

\(\displaystyle 36 \div 5 = 7 \frac{1}{5}\) inches.

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

Example Question #122 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

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:

\(\displaystyle 25\; \textrm{in}\)

\(\displaystyle 20 \; \textrm{in}\)

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

\(\displaystyle 30 \; \textrm{in}\)

\(\displaystyle 15\; \textrm{in}\)

Correct answer:

\(\displaystyle 20 \; \textrm{in}\)

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

\(\displaystyle 12 \div 3 = 4\) inches. 

The perimeter is 

\(\displaystyle 4 \times 5 = 20\) inches.

Example Question #1 : 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:

(A) is greater

(B) 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 \(\displaystyle s\) be the length of one side of the hexagon. Then its perimeter is \(\displaystyle 6s\).

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

\(\displaystyle s + 0.20s = 1.20 s\).

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

The perimeters are the same.

Example Question #5 : 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:

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

\(\displaystyle 120 \%\)

\(\displaystyle 92 \%\)

\(\displaystyle 111 \%\)

\(\displaystyle 96 \%\)

Correct answer:

\(\displaystyle 96 \%\)

Explanation:

Let \(\displaystyle s\) be the length of one side of the regular pentagon. Then its perimeter is \(\displaystyle 5s\).

The length of one side of the regular octagon is 60% of \(\displaystyle s\), or \(\displaystyle 0.6s\), so its perimeter is \(\displaystyle 8 \cdot 0.6s = 4.8 s\).The answer is therefore the percent \(\displaystyle 4.8s\) is of \(\displaystyle 5s\), which is

\(\displaystyle \frac{4.8s}{5s} \times 100 = 0.96 \times 100 = 96 \%\)

Example Question #6 : 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) and (B) are equal

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

(A) is greater

(B) is greater

Correct answer:

(A) is greater

Explanation:

Let \(\displaystyle s\) be the length of one side of the pentagon. Then its perimeter is \(\displaystyle 5s\).

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

\(\displaystyle s - 0.20s = 0.80 s\).

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

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

Example Question #7 : Pentagons

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

Which quantity is greater?

(a) \(\displaystyle x + y\)

(b) 180

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:

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

\(\displaystyle x + x + x + y + 2y = 540\)

\(\displaystyle 3x + 3y = 540\)

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

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

\(\displaystyle x + y = 180\)

making the two quantities equal.

Example Question #8 : Pentagons

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

Which quantity is greater?

(a) \(\displaystyle x\)

(b) \(\displaystyle y\)

Possible Answers:

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Correct answer:

It is impossible to tell from the information given.

Explanation:

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

\(\displaystyle x + x + x + y + y = 540\)

\(\displaystyle 3x + 2y = 540\)

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

Example Question #9 : Pentagons

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

Which is the greater quantity?

(a) \(\displaystyle m \angle BAC\)

(b) \(\displaystyle m \angle VUW\)

Possible Answers:

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

(a) 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,  \(\displaystyle m \angle BAC = m \angle BCA\) and  \(\displaystyle m \angle VUW = m \angle VWU\). Also, since the measures of the angles of a triangle total \(\displaystyle 180^{\circ }\), we know that 

\(\displaystyle m \angle BAC + m \angle BCA + m \angle ABC = 180\)

and 

\(\displaystyle m \angle VWU + m \angle VUW + m \angle UVW = 180\).

We can use these equations to compare \(\displaystyle m \angle BAC\) and \(\displaystyle m \angle VUW\).

 

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

\(\displaystyle m \angle BAC + m \angle BCA + m \angle ABC = 180 ^{\circ }\)

\(\displaystyle m \angle BAC + m \angle BAC + 108 ^{\circ } = 180 ^{\circ }\)

\(\displaystyle 2\cdot m \angle BAC + 108 ^{\circ }= 180 ^{\circ }\)

\(\displaystyle 2\cdot m \angle BAC + 108 ^{\circ } -108 ^{\circ } = 180 ^{\circ }-108 ^{\circ }\)

\(\displaystyle 2\cdot m \angle BAC = 72 ^{\circ }\)

\(\displaystyle 2 \cdot m \angle BAC \div 2 = 72 ^{\circ } \div 2\)

\(\displaystyle m \angle BAC = 36 ^{\circ }\)

 

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

\(\displaystyle m \angle VWU + m \angle VUW + m \angle UVW = 180 ^{\circ }\)

\(\displaystyle m \angle VUW + m \angle VUW + 120^{\circ } = 180 ^{\circ }\)

\(\displaystyle 2 \cdot m \angle VUW + 120^{\circ } = 180 ^{\circ }\)

\(\displaystyle 2 \cdot m \angle VUW + 120^{\circ } -120^{\circ } = 180 ^{\circ }-120^{\circ }\)

\(\displaystyle 2 \cdot m \angle VUW = 60^{\circ }\)

\(\displaystyle 2 \cdot m \angle VUW \div 2 = 60^{\circ }\div 2\)

\(\displaystyle m \angle VUW = 30^{\circ }\)

 

\(\displaystyle m \angle BAC > m \angle VUW\)

 

 

 

Example Question #10 : Pentagons

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

Which is the greater quantity? 

(a) \(\displaystyle AC\)

(b) \(\displaystyle UW\)

Possible Answers:

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

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:

\(\displaystyle \overline{AB} \cong \overline{ UV }\)

\(\displaystyle \overline{BC} \cong \overline{ VW }\)

Their included angles, \(\displaystyle \angle B\) and \(\displaystyle \angle V\), 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:

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

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

\(\displaystyle m\angle V > m\angle B \Rightarrow UW > AC\)

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