Perimeter and Sides of Quadrilaterals - GED Math

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Question

The length of each side of a square is increased by 12%. By what percent has its perimeter increased?

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Answer

Let be the original sidelength of the square. Increasing this by 12% is the same as adding 0.12 times that sidelength to the original sidelength. The new sidelength is therefore

.

The perimeter of a square is four times its sidelength. Its original perimeter was ; after the increase, it is $4 \cdot 1.12s = 4.48 s$ . The percent of increase is therefore

$\frac{4.48s - 4s }{4s} \times 100 % = \frac{0.48 s}{4s} \times 100 % = 0.12 \times 100 % = 12 %$ .

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Question

Note: Figure NOT drawn to scale

Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in brown) six feet wide throughout. What is the perimeter of the garden?

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Answer

The inner rectangle, which represents the garden, has length and width $(100 - 2\times 6) = 88$ feet and $(60 - 2\times 6) = 48$ feet, respectively, so its perimeter is

feet.

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Question

Note: Figure NOT drawn to scale.

Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in brown). The dirt path is feet wide throughout. Which of the following polynomials gives the perimeter of the garden?

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Answer

The length of the garden is than that of the entire lot, or

.

The width of the garden is than that of the entire lot, or

.

The perimeter is twice the sum of the two:

$= 2 (100-2x+60-2x) = 2\left ( 160-4x\right )= 320-8x$

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Question

Note: Figure NOT drawn to scale

Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in brown). The dirt path is six feet wide throughout. Which of the following polynomials gives the perimeter of the garden, in feet?

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Answer

The length of the garden, in feet, is $2 \times 6 = 12$ feet less than that of the entire lot, or

.

The width of the garden, in feet, is $2 \times 6 = 12$ less than that of the entire lot, or

.

The perimeter, in feet, is twice the sum of the two:

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Question

Note: Figure NOT drawn to scale

Give the ratio of the perimeter of Rectangle to that of Rectangle .

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Answer

The perimeter of Rectangle is

.

Opposite sides of a rectangle are congruent, so

and

.

The perimeter of Rectangle is

.

Opposite sides of a rectangle are congruent, so

,

,

and

.

The ratio of the perimeters is

$\frac{60}{42} = \frac{60\div 6}{42\div 6} =\frac{10}{7}$ - that is, 10 to 7.

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Question

Given Square , the distance from to is one mile. Which of the following gives the perimeter of the square, to the nearest foot?

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Answer

The segment $\overline{AC}$ is a diagonal of the square. It is also the hypotenuse of a right triangle whose legs both measure the same length, which we will call ; this is also the length of one side of the square.

The right triangle formed is an isosceles triangle - and, subsequently, a $45^{\circ }-45^{\circ }-90^{\circ }$ triangle - and therefore,

$s \sqrt{2} = AC$

$\overline{AC}$ is one mile, or 5,280 feet, long, so we substitute and solve for :

$s \sqrt{2} = 5,280$

$s = 5,280 \div \sqrt{2} \approx 5,280 \div 1.414 \approx 3,733.5$

The perimeter, in feet, is four times this, or

$P = 4s \approx 4 \times 3,733.5 \approx 14,934$ feet.

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Question

Which of the following can be the sidelengths of a rhombus?

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Answer

The four sides of a rhombus have equal length, so we can eliminate three choices by demonstrating that at least two sidelengths are not equal.

$0.1\textrm{ km}, \textup{1,000}\textrm{ m}, \textup{1,000}\textrm{ m}, 1\textrm{ hm}$ :

1,000 meters is, by definition, equal to 1 kilometer, not 0.1 kilometers. Therefore,

$0.1\textrm{ km} \neq \textup{1,000}\textrm{ m}$

and this choice is incorrect.

$1\textrm{ mi}, \textrm{1 mi}, 1,760 \textrm{ ft}, 586 \frac{2}{3} \textrm{ yd}$ :

1 mile is, by definition, equal to 5,280 feet, not 1,760 feet. Therefore,

$1\textrm{ mi} \neq 1,760$

and this choice is incorrect.

$10 \textrm{ cm}, 1\textrm{ dm}, 0.01 \textrm{ m}, 0.01 \textrm{ m}$

By definition, 1 decimeter, not 0.1 decimeter, is equal to 1 meter. Therefore,

$1 \textrm{ dm} \neq 0.01 \textrm{ m}$

and this choice is incorrect.

$18\textrm{ in}, 18\textrm{ in}, 1\frac{1}{2} \textrm{ ft}, \frac{1}{2} \textrm{ yd}$ :

$\frac{1}{2}$ yard is equal to $\frac{1}{2} \times 36 = 18$ inches and, also, $\frac{1}{2} \times 3 = 1\frac{1}{2}$ feet. Therefore,

$18\textrm{ in} = 18\textrm{ in} = 1\frac{1}{2} \textrm{ ft}= \frac{1}{2} \textrm{ yd}$

All four sides have equal length so this is the rhombus. This is the correct choice.

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Question

Identify the above polygon.

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Answer

A polygon with six sides is called a hexagon.

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Question

Refer to the above figure. You are given that $\overleftrightarrow{AE} ||\overleftrightarrow{BD}$ and that $\angle FCD$ is acute.

Which of the following words accurately describes Polygon ?

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Answer

Polygon has four sides and is therefore a quadrilateral. $\overleftrightarrow{AE} ||\overleftrightarrow{BD}$ , so $\overline{FE }|| \overline{CD}$ . Also, since $\angle FCD$ is acute and $\angle CDE$ is right, $\angle FCD \ncong \angle CDE$ , so $\overline{FC } \nparallel \overline{DE}$ .

The quadrilateral has one pair of parallel sides, and the other two sides are not parallel. Therefore, it is a trapezoid.

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Question

Refer to the above three figures. All parallel sides are so indicated.

Which of the figures can be called a quadrilateral?

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Answer

By definition, any polygon with four sides is called a quadrilateral. All three figures fit this description.

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Question

Refer to the above three figures. All parallel sides are so indicated.

Which of the figures can be called a parallelogram?

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Answer

A parallelogram, by definition, has two pairs of parallel sides. Figures A and B fit that criterion, but Figure C does not.

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Question

Refer to the above diagram. Parallel sides are so indicated.

Identify the above polygon.

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Answer

A four-sided figure, or quadrilateral, with one pair of parallel sides and its other sides nonparallel is called a trapezoid.

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Question

Note: Figure NOT drawn to scale.

Quadrilateral is a rhombus. Calculate its perimeter if:

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Answer

The four sides of a rhombus are congruent. Also, the diagonals of a rhombus are perpendicular bisectors to each other, so the four triangles they form are right triangles. Therefore, the Pythagorean theorem can be used to determine the common sidelength of Quadrilateral .

We focus on $\Delta AXB$ . The diagonals of a rhombus, as is the case with any parallelogram, are each the other's bisector, so

$AX = \frac{1}{2} (AC) = \frac{1}{2} \cdot 24 = 12$

$XB = \frac{1}{2} (BD) = \frac{1}{2} \cdot 10 = 5$

By the Pythagorean Theorem,

$AB = \sqrt{(AX)^{2}+ (XB)^{2}} = \sqrt{12^{2}+5^{2}} = \sqrt{144+25} = \sqrt{169} = 13$

13 is the common length of the four sides of Quadrilateral , so its perimeter is $4 \times 13 = 52$ .

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Question

The perimeter of a square is 24cm. Find the length of one side of the square.

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Answer

The formula to find the perimeter of a square is

where a is the length of any side of the square. Because a square has 4 equal sides, we can use any side in the formula. To find the length of one side of the square, we will solve for a.

Now, we know the perimeter of the square is 24cm. So, we will substitute and solve for a. We get

$24\text{cm} = 4a$

$\frac{24\text{cm}}{4} = \frac{4a}{4}$

$6\text{cm} = a$

$a = 6\text{cm}$

Therefore, the length of one side of the square is 6cm.

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Question

Find the perimeter of a rectangle with a width of 8in and a length that is two times the width.

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Answer

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

Where l is the length and w is the width of the rectangle.

Now, we know the width of the rectangle is 8in. We also know the length is two times the width. Therefore, the length is 16in. So, we can substitute. We get

$P = 2(16\text{in}) + 2(8\text{in})$

$P = 32\text{in} + 16\text{in}$

$P = 48\text{in}$

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Question

Find the perimeter of a square with a length of 8cm.

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Answer

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

where a is the length of one side of the square. Because a square has 4 equal sides, we can use any side in the formula.

Now, we know the length of the square is 8cm. So, we can substitute. We get

$P = 4 \cdot 8\text{cm}$

$P = 32\text{cm}$

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Question

The perimeter of a square is 72in. Find the length of one side.

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Answer

To answer this, we will look at the formula for perimeter of a square. We get

where a is the length of one side of the square. Because a square has 4 equal sides, we can use any side in the formula. To find the length of one side, we will solve for a. Now, we know the perimeter of the square is 72\text{in}. So, we will substitute and solve for a. We get

$72\text{in} = 4a$

$\frac{72\text{in}}{4} = \frac{4a}{4}$

$18\text{in} = a$

$a = 18\text{in}$

Therefore, the length of one side of the square is 18in.

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Question

The area of a square is $25\text{cm}^2$ . Find the perimeter.

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Answer

A square has 4 equal sides. To find the perimeter of the square, we first have to find the length of one side of the square.

Now, we know the area of the square. The formula for area of a square is

where b is the length of one side of the square. To find the length of one side, we will solve for b. Now, we know the area is $25\text{cm}^2$ . So, we will substitute and solve for b.

$25\text{cm}^2 = b^2$

$\sqrt{25\text{cm}^2} = \sqrt{b^2}$

$5\text{cm} = b$

$b = 5\text{cm}$

Therefore, the length of one side of the square is 5cm.

Now, to find the perimeter of a square, we will use the following formula:

where b is the length of one side of the square. Now, we know the length of one side of the square is 5cm. So, we can substitute. We get

$P = 4 \cdot 5\text{cm}$

$P = 20\text{cm}$

Therefore, the perimeter of the square is 20cm.

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Question

Find the perimeter of a rectangle with a length of 9in and a width that is a third of the length.

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Answer

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

where l is the length and w is the width of the rectangle.

Now, we know the length of the rectangle is 9in. We also know the width is a third of the length. Therefore, the width is 3in. So, we can substitute. We get

$P = 2(9\text{in}) + 2(3\text{in})$

$P = 18\text{in} + 6\text{in}$

$P = 24\text{in}$

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Question

The perimeter of a square is 68in. Find the length of one side.

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Answer

A square has 4 equal sides. To find the perimeter of a square, we will use the following formula:

where a is the length of one side of the square. Because a square has 4 equal sides, we can use any side in the formula. To find the length of one side, we will solve for a.

Now, we know the perimeter of the square is 68in. So, we can substitute and solve for a.

$68\text{in} = 4a$

$\frac{68\text{in}}{4} = \frac{4a}{4}$

$17\text{in} = a$

$a = 17\text{in}$

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

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