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

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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:

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

Correct answer:

(b) is greater.

Explanation:

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

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

$\displaystyle 5x + 1,000 = 5,280$

$\displaystyle 5x + 1,000 - 1,000 = 5,280- 1,000$

$\displaystyle 5x = 4,280$

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

x = 856 $\displaystyle x = 856$ 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 $856 \times 2 = 1,712$ $\displaystyle 856 \times 2 = 1,712$ 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) is greater

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

(B) is greater

(A) and (B) are equal

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

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

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

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All ISEE Upper Level Quantitative Resources

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

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

All ISEE Upper Level Quantitative Resources

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