Squares, Rectangles, and Parallelograms - GED Math

Card 0 of 505

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?

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

Your mother made a casserole in a pan whose length is 9 inches and whose width is one and a half times the length. What is the area of the casserole pan?

Answer

To find the area, you need to multiply the dimensions. First, however, we need to find our width.

We can find our width by using the clue: "...one and a half times the length"

So, do the following:

Next, perform the following operation:

Making our answer:

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Question

Find the area of a square if the side length is $\sqrt{24}$ .

Answer

The area of a square is:

Substitute the side into the area formula of the square.

$A =(\sqrt{24})^2$

Squaring a radical will leave just the term inside the radical.

The area is:

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Question

A rectangle has a diagonal of and a length of . What is the area of the rectangle?

Answer

The rectangle given in the question can be drawn out as thus:

Notice that the diagonal is also the hypotenuse of a right triangle that has the length and width of the rectangle as its legs. Thus, use the Pythagorean Theorem to find the length of the width of the rectangle.

$8^2+\text{width}^2=12^2$

$64+\text{width}^2=144$

$\text{width}^2=80$

$\text{width}=8.94$

Next, recall how to find the area of a rectangle.

$\text{Area of Rectangle}=\text{length}\times \text{width}$

Plug in the length and width of the rectangle.

$\text{Area }=8 \times 8.94 =71.52$

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Question

Suppose you receive a square sheet of wood which you are planning on making into a box. If the sheet is , find the perimeter of the sheet.

Answer

Suppose you receive a square sheet of wood which you are planning on making into a box. If the sheet is , find the perimeter of the sheet.

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

So, let's plug in our length and width and solve:

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Question

Use the following rectangle to answer the question:

Find the perimeter.

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, given the rectangle

we can see the length is 9cm and the width is 7cm. So, we can substitute. We get

$P = 2(9\text{cm}) + 2(7\text{cm})$

$P = 18\text{cm} + 14\text{cm}$

$P = 32\text{cm}$

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Question

What is the perimeter of a square if the side length is ?

Answer

A square has four congruent sides.

Multiply the quantity by four.

The answer is:

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Question

A rectangular television has a diagonal of inches and a width of inches. What is the area of the television?

Answer

The figure above represents the television.

Use the Pythagorean Theorem to find the length of the rectangle.

$40^2+\text{length}^2=48^2$

$\text{length}^2=704$

$\text{length}=\sqrt{704}$

Next, recall how to find the area of a rectangle.

$\text{Area of Rectangle}=\text{length}\times \text{width}$

For the given rectangle,

$\text{Area}=40(\sqrt{704})=1061.32$

Make sure to round to two places after the decimal.

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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?

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?

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

In Rhombus , $m \angle R = 60 ^{\circ }$ . If $\overline{RO }$ is constructed, which of the following is true about $\Delta RHO$ ?

Answer

The figure referenced is below.

The sides of a rhombus are congruent by definition, so $\overline{RH} \cong \overline{HO}$ , making $\Delta RHO$ isosceles (and possibly equilateral).

Also, consecutive angles of a rhombus are supplementary, as they are with all parallelograms, so

$m \angle H = 180^{\circ} - m \angle HRM = 180^{\circ} - 60 ^{\circ} = 120^{\circ}$ .

$\angle H$ , having measure greater than $90^{\circ}$ , is obtuse, making $\Delta RHO$ an obtuse triangle. Also, the triangle is not equilateral, since such a triangle must have three $60 ^{\circ}$ angles.

The correct response is that $\Delta RHO$ is obtuse and isosceles, but not equilateral.

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Question

Given Quadrilateral , which of these statements would prove that it is a parallelogram?

I) $\angle A \cong \angle B$ and $\angle C \cong \angle D$

II) $\angle A \cong \angle C$ and $\angle B \cong \angle D$

III) $\angle A$ and $\angle B$ are supplementary and $\angle C$ and $\angle D$ are supplementary

Answer

Statement I asserts that two pairs of consecutive angles are congruent. This does not prove that the figure is a parallelogram. For example, an isosceles trapezoid has two pairs of congruent base angles, which are consecutive.

Statement II asserts that both pairs of opposite angles are congruent. By a theorem of geometry, this proves the quadrilateral to be a parallelogram.

Statement III asserts that two pairs of consecutive angles are supplementary. While all parallelograms have this characteristic, trapezoids do as well, so this does not prove the figure a parallelogram.

The correct response is Statement II only.

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Question

You are given Parallelogram with . Which of the following statements, along with what you are given, would be enough to prove that Parallelogram is a rectangle?

I) $m \angle A = 90 ^{\circ }$

II) $m \angle B = 90 ^{\circ }$

III) $m \angle C = 90 ^{\circ }$

Answer

A rectangle is defined as a parallelogram with four right, or $90 ^{\circ}$ , angles.

Since opposite angles of a paralellogram are congruent, if one angle measures $90 ^{\circ}$ , so does its opposite. Since consecutive angles of a paralellogram are supplementary - that is, their degree measures total $180 ^{\circ}$ - if one angle measures $90 ^{\circ}$ , then both of the neighboring angles measure $180 ^{\circ} - 90 ^{\circ} = 90 ^{\circ}$ .

In short, in a parallelogram, if one angle is right, all are right and the parallelogram is a rectangle. All three statements assert that one angle is right, so from any one, it follows that the figure is a rectangle. The correct response is Statements I, II, or III.

Note that the sidelengths are irrelevant.

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Question

If the rectangle has a width of 5 and a length of 10, what is the area of the rectangle?

Answer

Write the area for a rectangle.

Substitute the given dimensions.

The answer is:

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Question

In the figure below, find the measure of the largest angle.

Answer

Recall that in a quadrilateral, the interior angles must add up to $360^{\circ}$ .

Thus, we can solve for :

Now, to find the largest angle, plug in the value of into each expression for each angle.

The largest angle is $151^{\circ}$ .

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Question

Find the area of the trapezoid:

Answer

The area of a trapezoid is calculated using the following equation:

$A=\frac{b_1+b_2}{2}h=\frac{7+5}{2}(3)=\frac{12}{2}(3)=(6)(3)=18$

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Question

A rectangle has length 10 inches and width 5 inches. Each dimension is increased by 3 inches. By what percent has the area of the rectangle increased?

Answer

The area of a rectangle is its length times its width.

Its original area is $10 \times 5 = 50$ square inches; its new area is $13 \times 8 = 104$ square inches. The area has increased by

$\frac{104 - 50}{50 } \times 100 = \frac{54}{50 } \times 100 = 108 %$ .

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Question

A rectangle has length 10 inches and width 8 inches. Its length is increased by 2 inches, and its width is decreased by 2 inches. By what percent has the area of the rectangle decreased?

Answer

The area of a rectangle is its length times its width.

Its original area is $10 \times 8 = 80$ square inches; its new area is $12 \times 6 = 72$ square inches. The area has decreased by

$\frac{80 -72}{80} \times 100 % = \frac{8}{80} \times 100 % = 10 %$ .

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Question

The length of each side of a square is increased by 10%. By what percent has its area increased?

Answer

Let be the original sidelength of the square. Increasing this by 10% is the same as adding 0.1 times that sidelength to the original sidelength. The new sidelength is therefore

The area of a square is the square of its sidelength.

The area of the square was originally $s^{2}$ ; it is now

$\left ( 1.1s\right )^{2}= 1.21s^{2}$

That is, the area has increased by

$\frac{1.21s^{2}-s^{2}}{s^{2}} \times 100 % =\frac{0.21s^{2}}{s^{2}} \times 100 % = 0.21 \times 100 % = 21 %$ .

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Question

The above figure depicts the rectangular swimming pool at an apartment. The apartment manager needs to purchase a tarp that will cover this pool completely. However, because of the cutting device the pool store uses, the length and the width of the tarp must each be a multiple of four yards. Also, the tarp must be at least one yard longer and one yard wider than the pool.

What will be the minimum area of the tarp the manager purchases?

Answer

Three feet make a yard, so the length and width of the pool are $\frac{50}{3} = 16 \frac{2}{3}$ yards and $\frac{35}{3} = 11 \frac{2}{3}$ yards, respectively. Since the dimensions of the tarp must exceed those of the pool by at least one yard, the tarp must be at least $17 \frac{2}{3}$ yards by $12 \frac{2}{3}$ yards; but since both dimensions must be multiples of four yards, we take the next multiple of four for each.

The tarp must be 20 yards by 16 yards, so the area of the tarp is the product of these dimensions, or

$20 \times 16 = 320$ square yards.

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