GED Math : Geometry and Graphs

Study concepts, example questions & explanations for GED Math

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Example Questions

Example Question #42 : Squares, Rectangles, And Parallelograms

Garden

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 \(\displaystyle y\) feet wide throughout. Which of the following polynomials gives the area of the garden?

Possible Answers:

\(\displaystyle 320-8x\)

\(\displaystyle x^{2} - 160x +6,000\)

\(\displaystyle 320- 4x\)

\(\displaystyle 4y^{2} - 320 y + 6,000\)

Correct answer:

\(\displaystyle 4y^{2} - 320 y + 6,000\)

Explanation:

The length of the garden is \(\displaystyle 2y\) feet less than that of the entire lot, or 

\(\displaystyle L = 100 - 2y\).

The width of the garden is \(\displaystyle 2y\)  feet less than that of the entire lot, or 

\(\displaystyle W = 60 - 2y\).

The area of the garden is their product:

\(\displaystyle A =LW = (100-2y)(60-2y)\)

\(\displaystyle = 100 \cdot 60 - 100 \cdot 2y - 2y \cdot 60+2y \cdot 2y\)

\(\displaystyle = 6,000-200y-120y+4y^{2}\)

\(\displaystyle =4y^{2} - 320 y + 6,000\)

Example Question #341 : Geometry And Graphs

Rectangle

Note: Figure NOT drawn to scale

Refer to the above diagram. 

Half of Rectangle \(\displaystyle ACGF\) is pink. \(\displaystyle AB = 20, AD = 15, BC = 10\).

Evaluate \(\displaystyle DF\).

Possible Answers:

\(\displaystyle DF = 5\)

\(\displaystyle DF = 10\)

\(\displaystyle DF = 3\)

\(\displaystyle DF=9\)

Correct answer:

\(\displaystyle DF = 5\)

Explanation:

Rectangle \(\displaystyle ABED\) has length \(\displaystyle L = AB = 20\) and width \(\displaystyle W = AD = 15\), so it has area

\(\displaystyle LW = 20 \times 15 = 300\).

The area of Rectangle \(\displaystyle ACGF\) is twice that of Rectangle \(\displaystyle ABED\), or 600. Its length is 

\(\displaystyle L = AC = AB + BC = 20 + 10 = 30\).

Its width is 

\(\displaystyle W = AD + DF = 15 + DF\).

Plug in what we know and solve for \(\displaystyle DF\):

\(\displaystyle A = LW\)

\(\displaystyle 600 =30 \times \left (15 + DF \right )\)

\(\displaystyle 20 = 15 + DF\)

\(\displaystyle DF = 5\)

 

Example Question #51 : Squares, Rectangles, And Parallelograms

Garden

Note:  Figure NOT drawn to scale

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

Possible Answers:

\(\displaystyle 1,550\textrm{ ft}^{2}\)

\(\displaystyle 1,600\textrm{ ft}^{2}\)

\(\displaystyle 1,500\textrm{ ft}^{2}\)

\(\displaystyle 1,400\textrm{ ft}^{2}\)

Correct answer:

\(\displaystyle 1,500\textrm{ ft}^{2}\)

Explanation:

The dirt path can be seen as the region between two rectangles. The outer rectangle has length and width 100 feet and 60 feet, respectively, so its area is 

\(\displaystyle A = 100 \cdot 60 = 6,000\) square feet.

 

The inner rectangle has length and width \(\displaystyle (100 - 2\times 5) = 90\) feet and \(\displaystyle (60 - 2\times 5) = 50\) feet, respectively, so its area is

\(\displaystyle A = 90 \times 50 = 4,500\) square feet.

 

The area of the path is the difference of the two:

\(\displaystyle 6,000 - 4,500 = 1,500\) square feet.

Example Question #1 : Area Of A Quadrilateral

Rectangle

Note: Figure NOT drawn to scale

\(\displaystyle AB = 14, BC = 6, AD = 8, DF = 4\)

What percent of Rectangle \(\displaystyle ACGF\) is pink?

Possible Answers:

\(\displaystyle 44 \frac{4}{9} \%\)

\(\displaystyle 55 \frac{5}{9} \%\)

\(\displaystyle 46\frac{2}{3} \%\)

\(\displaystyle 50 \%\)

Correct answer:

\(\displaystyle 46\frac{2}{3} \%\)

Explanation:

The pink region is Rectangle \(\displaystyle ABED\). Its length and width are

\(\displaystyle L = AB = 14\)

\(\displaystyle W = AD = 8\)

so its area is the product of these, or

\(\displaystyle 14 \times 8 = 112\).

The length and width of Rectangle \(\displaystyle ACGF\) are

\(\displaystyle L = AC = AB + BC = 14 + 6 = 20\)

\(\displaystyle W =AF = AD + DF= 8 + 4 = 12\)

so its area is the product of these, or

\(\displaystyle 20 \times 12 = 240\).

So we want to know what percent 112 is of 240, which can be answered as follows:

\(\displaystyle \frac{112}{240} \times 100 \% = 46\frac{2}{3} \%\)

Example Question #1301 : Ged Math

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?

Possible Answers:

\(\displaystyle 10 \%\)

\(\displaystyle 11 \frac{1}{9} \%\)

\(\displaystyle 8 \%\)

\(\displaystyle 12 \frac{1}{2} \%\)

Correct answer:

\(\displaystyle 10 \%\)

Explanation:

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

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

\(\displaystyle \frac{80 -72}{80} \times 100 \% = \frac{8}{80} \times 100 \% = 10 \%\).

Example Question #51 : Squares, Rectangles, And Parallelograms

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?

Possible Answers:

\(\displaystyle 108 \%\)

\(\displaystyle 104 \%\)

\(\displaystyle 54 \%\)

\(\displaystyle 58 \%\)

Correct answer:

\(\displaystyle 108 \%\)

Explanation:

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

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

\(\displaystyle \frac{104 - 50}{50 } \times 100 = \frac{54}{50 } \times 100 = 108 \%\).

Example Question #343 : 2 Dimensional Geometry

Find the area of a square with a side of \(\displaystyle 4x^2\).

Possible Answers:

\(\displaystyle 8\pi x^2\)

\(\displaystyle 16 x^4\)

\(\displaystyle 16\pi^2 x^4\)

\(\displaystyle 8\pi^2 x^2\)

\(\displaystyle 8\pi x^4\)

Correct answer:

\(\displaystyle 16 x^4\)

Explanation:

Write the formula for the area of a square.

\(\displaystyle A=s^2\)

Substitute the side into the equation.

\(\displaystyle A= (4x^2)^2\)

Simplify the equation.

The answer is:  \(\displaystyle 16 x^4\)

Example Question #54 : Squares, Rectangles, And Parallelograms

If a rectangle has a length of 18cm and a width that is half the length, what is the area of the rectangle?

Possible Answers:

\(\displaystyle 144\text{cm}^2\)

\(\displaystyle 36\text{cm}^2\)

\(\displaystyle 162\text{cm}^2\)

\(\displaystyle 72\text{cm}^2\)

\(\displaystyle 126\text{cm}^2\)

Correct answer:

\(\displaystyle 162\text{cm}^2\)

Explanation:

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

\(\displaystyle A = l \cdot w\)

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

 

Now, we know the length of the rectangle is 18cm. We also know the width is half the length. Therefore, the width is 9cm. So, we can substitute.  We get

\(\displaystyle A = 18\text{cm} \cdot 9\text{cm}\)

\(\displaystyle A = 162\text{cm}^2\)

Example Question #13 : Area Of A Quadrilateral

If a square has a length of 10in, find the area.

Possible Answers:

\(\displaystyle 60\text{in}^2\)

\(\displaystyle 50\text{in}^2\)

\(\displaystyle 125\text{in}^2\)

\(\displaystyle 100\text{in}^2\)

\(\displaystyle 40\text{in}^2\)

Correct answer:

\(\displaystyle 100\text{in}^2\)

Explanation:

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

\(\displaystyle A = l \cdot w\)

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

Now, we know the length of the square is 10in. Because it is a square, all sides are equal. Therefore, the length is also 10in. So, we can substitute. We get

\(\displaystyle A = 10\text{in} \cdot 10\text{in}\)

\(\displaystyle A = 100\text{in}^2\)

Example Question #56 : Squares, Rectangles, And Parallelograms

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

Possible Answers:

\(\displaystyle 144\text{in}^2\)

\(\displaystyle 128\text{in}^2\)

\(\displaystyle 16\text{in}^2\)

\(\displaystyle 64\text{in}^2\)

\(\displaystyle 36\text{in}^2\)

Correct answer:

\(\displaystyle 128\text{in}^2\)

Explanation:

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

\(\displaystyle A = l \cdot w\)

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

\(\displaystyle A = 16\text{in} \cdot 8\text{in}\)

\(\displaystyle A = 128\text{in}^2\)

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