How to find perimeter - ISEE Middle Level Quantitative Reasoning

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Question

Which is the greater quantity?

(a) The perimeter of an equilateral triangle with sidelength 30 inches

(b) The perimeter of a square with sidelength 2 feet

Answer

Each figure has sides that are congruent, so in each case, multiply the sidelength by the number of sides.

(a) The triangle has perimeter $30 \times 3 = 90$ inches

(b) 2 feet are equal to 24 inches, so the square has sidelength $24 \times 4 = 96$ inches.

The square has the greater perimeter.

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Question

Which is the greater quantity?

(a) The perimeter of a right triangle with legs of length 3 inches and 4 inches

(b) One foot

Answer

The length of the hypotenuse of a triangle with legs 3 inches and 4 inches long is calculated using the Pythagorean Theorem, setting :

$c = \sqrt{a^{2}+b^{2}} = \sqrt{3^{2}+4^{2}} = \sqrt{9+16} = \sqrt{25} =5$

The hypotenuse is 5 inches long. The perimeter is therefore inches, which is equal to one foot.

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Question

Which is the greater quantity?

(a) The perimeter of a right triangle with legs of length 5 feet and 12 feet

(b) 8 yards

Answer

The length of the hypotenuse of a triangle with legs 5 feet and 12 feet is calculated using the Pythagorean Theorem, setting :

$c = \sqrt{a^{2}+b^{2}} = \sqrt{5^{2}+12^{2}} = \sqrt{25+144} = \sqrt{169} =13$

The hypotenuse is 13 feet long. The perimeter is feet, which is equal to 10 yards.

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Question

Which is the greater quantity?

(a) The perimeter of a right triangle with hypotenuse of length 25 centimeters and one leg of length 7 centimeters

(b) One-half of a meter

Answer

The length of the second leg of the triangle can be calculated using the Pythagorean Theorem, setting :

$b = \sqrt{c^{2}-a^{2}} = \sqrt{25^{2}-7^{2}} = \sqrt{625-49}= \sqrt{576} = 24$

The second leg has length 24 centimeters, so the perimeter of the triangle is

centimeters.

One-half of a meter is one-half of 100 centimeters, or 50 centimeters, so (a) is greater.

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Question

$\Delta ABC$ is an equilateral triangle; .

Rectangle ;

Which is the greater quantity?

(a) The perimeter of $\Delta ABC$

(b) The perimeter of Rectangle

Answer

(a) The perimeter of the equilateral triangle is $25 \times 3 = 75$ .

(b) ; are of unknown value, but they are equal, so we will call their common length .

Rectangle has perimeter

.

Without knowing , it cannot be determined with certainty which figure has the longer perimeter. For example:

If , then $70 + 2N = 70 + 2 \cdot 1 = 70 + 2 = 72 < 75$

If , then $70 + 2N = 70 + 2 \cdot 3 = 70 + 6 = 76>75$

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Question

$\Delta ABC$ is an isosceles triangle; $\Delta DEF$ is an equilateral triangle

Which is the greater quantity?

(a) The perimeter of $\Delta ABC$

(b) The perimeter of $\Delta DEF$

Answer

(a) As an isosceles triangle, $\Delta ABC$ , by definition, has two congruent sides. $AB \neq BC$ , so either :

in which case the perimeter of $\Delta ABC$ is

or

in which case the perimeter of $\Delta ABC$ is

(b) $\Delta DEF$ is an equilateral triangle, so, by definition, all of its sides are congruent; its perimeter is $35 \times 3 = 105$ .

Regardless of the length of $\overline{AC}$ , $\Delta ABC$ has the greater perimeter.

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Question

$\Delta ABC$ and $\Delta DEF$ are right triangles, with right angles $\angle B , \angle E$ , respectively.

Which is the greater quantity?

(a)

(b)

Answer

(a) $\overline{AC}$ is the hypotenuse of $\Delta ABC$ , so by the Pythagorean Theorem,

$\left (AC \right )^{2} =\left ( AB \right ) ^{2} + \left ( B C\right )^{2}$

$\left (AC \right )^{2} =6^{2} + 8^{2}$

$\left (AC \right )^{2} =36 + 64 = 100$

$AC = \sqrt{100} = 10$

(b) $\overline{EF}$ is a leg of $\Delta DEF$ , whose hypotenuse is $\overline{DF}$ , so by the Pythagorean Theorem,

$\left (EF \right )^{2} =\left ( DF \right ) ^{2} - \left ( DE\right )^{2}$

$\left (EF \right )^{2} =26^{2} - 24^{2}$

$\left (EF \right )^{2} =676 -576 = 100$

$EF= \sqrt{100} = 10$

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Question

$\Delta ABC$ is a right triangle with hypotenuse $\overline{AC}$ 10 inches long.

The lengths of $\overline{AB}$ and $\overline{BC}$ , in inches, can both be expressed as integers.

Which is the greater quantity?

(a) $2 \textrm{ feet}$

(b) The perimeter of $\Delta ABC$

Answer

By the Pythagorean Theorem,

$(AB)^{2}+ (BC)^{2}= (AC)^{2}$

$(AB)^{2}+ (BC)^{2}= 10^{2}$

$(AB)^{2}+ (BC)^{2}= 100$

By trial and error, it can be determined that the only two positive integers that can replace and to make this equation true are 6 and 8, in either order:

$6^{2}+ 8^{2}= 100$

Add the three side lengths to get the perimeter inches, which is equal to 2 feet.

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Question

Note: Figure NOT drawn to scale.

Refer to the above triangle. An insect walks directly from A to B, then directly from B to C. What fraction of the perimeter of the triangle has he walked?

Answer

By the Pythagorean Theorem, the distance from A to C, which we will call , is equal to

$D = \sqrt{ 3^{2}+ 4^{2}} = \sqrt{ 9+ 16} = \sqrt{ 25} = 5$

The perimeter of the triangle is . The insect has traveled units, or $\frac{7}{12}$ of the perimeter.

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Question

An equilateral triangle has perimeter two yards. Which is the greater quantity?

(A) The length of one side of the triangle

(B) 28 inches

Answer

An equilateral triangle has three sides of equal length; the perimeter of this triangle is two yards, which is equal to $2 \times 36 = 72$ inches. One side has length $72 \div 3 = 24$ inches, which is less than 28 inches, so (B) is greater.

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Question

Note: Figure NOT drawn to scale.

Refer to the above triangle. An insect walks directly from B to A, then directly from A to C. What percent of the perimeter of the triangle has he walked?

Answer

By the Pythagorean Theorem, the distance from A to C, which we will call , is equal to

$D = \sqrt{ 3^{2}+ 4^{2}} = \sqrt{ 9+ 16} = \sqrt{ 25} = 5$

The perimeter of the triangle is . The insect has traveled units out of 12, which is

$\frac{8}{12} \times 100 = 66 \frac{2}{3} %$ of the perimeter.

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Question

Note: Figure NOT drawn to scale

Refer to the above triangle. An insect walks directly from B to C, then directly from C to A. What fraction of the perimeter of the triangle has he walked?

Answer

By the Pythagorean Theorem, the distance from B to C, which we will call , is equal to

$D = \sqrt{13^{2}- 5^{2}} = \sqrt{169- 25 }= \sqrt{144} = 12$

The perimeter of the triangle is

.

The insect has traveled units, which is

$\frac{25}{30} = \frac{25 \div 5}{30 \div 5} = \frac{5}{6}$

of the perimeter.

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Question

Note: Figure NOT drawn to scale

Refer to the above triangle. An insect walks directly from A to B, then directly from B to C. What percent of the perimeter of the triangle has he walked?

Answer

By the Pythagorean Theorem, the distance from B to C, which we will call , is

$D = \sqrt{13^{2}- 5^{2}} = \sqrt{169- 25 }= \sqrt{144} = 12$ .

The perimeter of the triangle is

.

The insect has traveled units, or

$\frac{17}{30} \times 100 = 56 \frac{2}{3} %$ of the perimeter.

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Question

What is the perimeter of a square with area 196 square inches?

Answer

A square with area 196 square inches has sidelength $\sqrt{196} = 14$ inches, and therefore has perimeter $4 \cdot 14 = 56$ inches

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Question

If a square has sides measuring $\small \small \frac{1}{8}$ , what is the perimeter of the square, in simplest form?

Answer

To find the perimeter of a square, you must add together all the sides. In this case, we are adding $\small \frac{1}{8}$ four times.

$\small \frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{4}{8}$

Since all of the denominators are the same, there is no need to find a commond denominator, so we add together the numerators. This gives us $\small \frac{4}{8}$ .

Since both the numerator and denomator are divisible by four, we must simplify this fraction.

$\small \frac{4/4}{8/4}= \frac{1}{2}$

The perimeter of the square is $\small \small \frac{1}{2}$ .

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Question

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

Answer

A square has four 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

$4 \times 4 = 16$ inches.

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Question

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

(A) The length of one side of the square

(B) 8 inches

Answer

One yard is equal to 36 inches. A square has four sides of equal length, so one side of the square has length

$36 \div 4 = 9$ inches.

Since , (A) is greater.

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Question

A square has perimeter five meters. Which is the greater quantity?

(A) 1,250 millimeters

(B) The length of one side of the square

Answer

One meter is equal to 1,000 millimeters, so the square has perimeter

$5 \times 1,000 = 5,000$ millimeters.

A square has four sides of equal length, so one side of the square has length

$5,000 \div 4 = 1,250$ millimeters.

The quantities are equal.

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Question

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

(A) 250 centimeters

(B) The length of one side of the square

Answer

One meter is equal to 100 centimeters.

A square has four sides of equal length, so we will need to divide by 4 to find the length of one side.

$P\div4=s$

$100cm \div 4 = 25cm$

, so (A) is greater

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Question

A square lot has perimeter one mile. Which is the greater quantity?

(A) 1,320 feet

(B) The length of one side of the square

Answer

One mile is equal to 5,280 feet. A square has four sides of equal length, so one side of the square has length

$5,280 \div 4 = 1,320$ feet.

The quantities are equal.

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