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ISEE Upper Level Quantitative : How to find the length of a side

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

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Example Questions

Example Question #4 : Other Polygons

Right_triangle

A regular decagon has the same perimeter as the above right triangle. Give the length of one side.

Possible Answers:

$14 \textrm{ in}$ $\displaystyle 14 \textrm{ in}$

$6 \frac{1}{5 }\textrm{ in}$ $\displaystyle 6 \frac{1}{5 }\textrm{ in}$

$11 \frac{1}{5} \textrm{ in}$ $\displaystyle 11 \frac{1}{5} \textrm{ in}$

$7 \frac{3}{4} \textrm{ in}$ $\displaystyle 7 \frac{3}{4} \textrm{ in}$

$9 \frac{1}{3} \textrm{ in}$ $\displaystyle 9 \frac{1}{3} \textrm{ in}$

Correct answer:

$11 \frac{1}{5} \textrm{ in}$ $\displaystyle 11 \frac{1}{5} \textrm{ in}$

Explanation:

By the Pythagorean Theorem, the hypotenuse of the right triangle is

$\sqrt{14^{2}+48^{2}} = \sqrt{196+2,304} = \sqrt{2,500} = 50$ $\displaystyle \sqrt{14^{2}+48^{2}} = \sqrt{196+2,304} = \sqrt{2,500} = 50$ inches, making its perimeter

$\displaystyle 14 + 48 + 50 =112$ inches.

A regular decagon has ten sides of equal length, so each side measures

$112 \div 10 = 11 \frac{1}{5}$ $\displaystyle 112 \div 10 = 11 \frac{1}{5}$ inches.

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

A regular octagon has perimeter one meter. Which is the greater quantity?

(A) The length of one side

(B) 125 millimeters

Possible Answers:

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

(A) is greater

(B) is greater

(A) and (B) are equal

Correct answer:

(A) and (B) are equal

Explanation:

A regular octagon has eight sides of equal length. The perimeter of this octagon is one meter, which is equal to 1,000 millimeters; each side, therefore, has length

$1,000 \div 8 = 125$ $\displaystyle 1,000 \div 8 = 125$ millimeters

making the quantities equal.

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

A regular pentagon has sidelength 72; the perimeter of a regular hexagon is 80% of that of the pentagon. Which is the greater quantity?

(A) The length of one side of the hexagon

(B) 50

Possible Answers:

(B) is greater

(A) and (B) are equal

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

(A) is greater

Correct answer:

(B) is greater

Explanation:

A regular pentagon has five sides of equal length; since one side is 72 units long, its perimeter is

$72 \times 5 = 360$ $\displaystyle 72 \times 5 = 360$ .

80% of this is

$0.8 \times 360 = 288$ $\displaystyle 0.8 \times 360 = 288$ ,

so this is the length of the hexagon, and, since all six sides are of equal length, one side measures

$288 \div 6 = 48$ $\displaystyle 288 \div 6 = 48$

(B) is greater.

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Example Question #4 : Other Polygons

A regular octagon has twice the perimeter of a regular pentagon. What is the ratio of the sidelength of the octagon to that of the pentagon?

Possible Answers:

$8\textrm{ to }5$ $\displaystyle 8\textrm{ to }5$

$4\textrm{ to } 5$ $\displaystyle 4\textrm{ to } 5$

$5\textrm{ to }4$ $\displaystyle 5\textrm{ to }4$

$5\textrm{ to }8$ $\displaystyle 5\textrm{ to }8$

$5\textrm{ to }2$ $\displaystyle 5\textrm{ to }2$

Correct answer:

$5\textrm{ to }4$ $\displaystyle 5\textrm{ to }4$

Explanation:

The solution is independent of the actual lengths, so we assume the pentagon has sidelength 1. Its perimeter is therefore 5. Subsequently, the octagon's perimeter is twice this, or 10, and its sidelength is one-eighth of this, or

$\frac{10}{8} = \frac{5}{4}$ $\displaystyle \frac{10}{8} = \frac{5}{4}$ .

The ratio of the sidelength of the octagon to that of the pentagon is

$\frac{5}{4} : 1$ $\displaystyle \frac{5}{4} : 1$ or 5 to 4.

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Example Question #3 : How To Find The Length Of A Side

A regular octagon has perimeter one mile.Which is the greater quantity?

(a) The length of one side

(b) 880 feet

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:

A regular octagon has eight sides of equal length. The perimeter, which is the sum of the lengths of these sides, is one mile, which is equal to 5,280 feet. Therefore, the length of one side is

$5, 280 \textrm{ ft} \div 8 = 660 \textrm{ ft}$ $\displaystyle 5, 280 \textrm{ ft} \div 8 = 660 \textrm{ ft}$ . This makes the length of a side less than 880 feet.

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : How to find the length of a side

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #4 : Other Polygons

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

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

Example Question #4 : Other Polygons

Example Question #3 : How To Find The Length Of A Side

All ISEE Upper Level Quantitative Resources

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