Pentagons - ISEE Upper Level Quantitative Reasoning

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Question

In a pentagon in which all the angles are equal and the perimeter is 35 inches, how many inches would be the sum of 2 of the sides?

Answer

In a pentagon in which all angles are equal, all sides will also be equal; therefore, the length of one side can be found by dividing the perimeter by 5 (the number of sides in a pentagon). Given that $\frac{35}{5}=7$ , the sum of 2 of the sides will be

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Question

If a pentagon has a perimeter of 85in, what is the length of one side of the pentagon?

Answer

A pentagon has 5 equal sides. We know the formula to find perimeter of a pentagon is

where a is the length of one side of the pentagon. Now, we know the total perimeter is 85in. Knowing this, we will substitute into the formula. We get

$85\text{in} = 5a$

$\frac{85\text{in}}{5} = \frac{5a}{5}$

$17\text{in} = a$

Therefore, the length of one side of the pentagon is 17in.

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Question

The measures of the angles of a pentagon are $x^{\circ }, x^{\circ }, x^{\circ }, y^{\circ }, y^{\circ }$ . If , what is ?

Answer

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

If , then

$3x + 2 \cdot 100 = 540$

$x = 340 \div 3$

$x = 113 \frac{1}{3}^{\circ}$

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Question

A convex pentagon has four angles that measure $150^{\circ }$ each. What is the measure of the fifth angle?

Answer

The angles of a pentagon measure a total of $180 (5-2) = 540 ^{\circ }$ . If we let the unknown angle measure be $x^{\circ }$ , then from this information:

Since an angle measure cannot be negative, this pentagon cannot exist.

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Question

The measures of the angles of a pentagon are $x ^{\circ }, x ^{\circ }, x ^{\circ }, y ^{\circ }, y ^{\circ }$ . If $x = 100 ^{\circ }$ , what is ?

Answer

The angles of a pentagon measure a total of $180 (5-2) = 540 ^{\circ }$ . From the information, we know:

If $x = 100 ^{\circ }$ , then the above becomes:

$3 \cdot 100 + 2y = 540$

$2y \div 2 = 240 \div 2$

$y = 120^{\circ}$

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Question

Solve for :

Answer

The sum of the interior angles of a pentagon can be determined by the following equation, where is the number of sides:

Therefore:

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Question

The measures of the angles of a pentagon are $x^{\circ }, x^{\circ }, x^{\circ }, y^{\circ }, y^{\circ }$ . If , what is ?

Answer

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

If , then

$3x + 2 \cdot 100 = 540$

$x = 340 \div 3$

$x = 113 \frac{1}{3}^{\circ}$

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Question

A convex pentagon has four angles that measure $150^{\circ }$ each. What is the measure of the fifth angle?

Answer

The angles of a pentagon measure a total of $180 (5-2) = 540 ^{\circ }$ . If we let the unknown angle measure be $x^{\circ }$ , then from this information:

Since an angle measure cannot be negative, this pentagon cannot exist.

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Question

The measures of the angles of a pentagon are $x ^{\circ }, x ^{\circ }, x ^{\circ }, y ^{\circ }, y ^{\circ }$ . If $x = 100 ^{\circ }$ , what is ?

Answer

The angles of a pentagon measure a total of $180 (5-2) = 540 ^{\circ }$ . From the information, we know:

If $x = 100 ^{\circ }$ , then the above becomes:

$3 \cdot 100 + 2y = 540$

$2y \div 2 = 240 \div 2$

$y = 120^{\circ}$

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Question

Solve for :

Answer

The sum of the interior angles of a pentagon can be determined by the following equation, where is the number of sides:

Therefore:

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Question

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

Answer

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

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

feet

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

(a) The longest side has measure 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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Question

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

(A) The length of one side

(B) 7 inches

Answer

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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Question

In a pentagon in which all the angles are equal and the perimeter is 35 inches, how many inches would be the sum of 2 of the sides?

Answer

In a pentagon in which all angles are equal, all sides will also be equal; therefore, the length of one side can be found by dividing the perimeter by 5 (the number of sides in a pentagon). Given that $\frac{35}{5}=7$ , the sum of 2 of the sides will be

Compare your answer with the correct one above

Question

If a pentagon has a perimeter of 85in, what is the length of one side of the pentagon?

Answer

A pentagon has 5 equal sides. We know the formula to find perimeter of a pentagon is

where a is the length of one side of the pentagon. Now, we know the total perimeter is 85in. Knowing this, we will substitute into the formula. We get

$85\text{in} = 5a$

$\frac{85\text{in}}{5} = \frac{5a}{5}$

$17\text{in} = a$

Therefore, the length of one side of the pentagon is 17in.

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Question

In a pentagon in which all the angles are equal and the perimeter is 35 inches, how many inches would be the sum of 2 of the sides?

Answer

In a pentagon in which all angles are equal, all sides will also be equal; therefore, the length of one side can be found by dividing the perimeter by 5 (the number of sides in a pentagon). Given that $\frac{35}{5}=7$ , the sum of 2 of the sides will be

Compare your answer with the correct one above

Question

If a pentagon has a perimeter of 85in, what is the length of one side of the pentagon?

Answer

A pentagon has 5 equal sides. We know the formula to find perimeter of a pentagon is

where a is the length of one side of the pentagon. Now, we know the total perimeter is 85in. Knowing this, we will substitute into the formula. We get

$85\text{in} = 5a$

$\frac{85\text{in}}{5} = \frac{5a}{5}$

$17\text{in} = a$

Therefore, the length of one side of the pentagon is 17in.

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Question

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

Answer

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

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

feet

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

(a) The longest side has measure 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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Question

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

(A) The length of one side

(B) 7 inches

Answer

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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Question

Find the perimeter of a pentagon with a side of length 12in.

Answer

To find the perimeter of a pentagon, we will use the following formula:

$P = 5 \cdot a$

where a is the length of one side of the pentagon.

Now, we know the length of one side of the pentagon is 12in.

Knowing this, we can substitute into the formula. We get

$P = 5 \cdot 12\text{in}$

$P = 60\text{in}$

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Question

Find the perimeter of a pentagon with a side having a length of 21in.

Answer

A pentagon has 5 sides. To find the perimeter, we will use the following formula:

where a is the length of a side of the pentagon.

Now, we know the length of a side of the pentagon is 21in.

Knowing this, we will substitute into the formula. We get

$P = 5 \cdot 21\text{in}$

$P = 105\text{in}$

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