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GED®

Prep Live:

The Coordinate Plane & Geometry

Learning Objectives

By the end of this lesson, you will be able to:

• Graph linear equations

• Graph linear inequalities

• Connect and interpret graphs and functions

• Connect coordinates, lines, and equations

• Calculate dimensions, perimeter, circumference, and area of two-dimensional figures

• Calculate dimensions, surface area, and volume of three-dimensional figures

Graphing Inequalities

When graphing an inequality on the number line, use an open circle to denote:

• “greater than” (>)

• “less than” (<)

Use a closed circle to denote:

• “greater than or equal to” (≥)

• “less than or equal to” (≤)

p. 358

2x

3x

1x

2x

A.

B.

C.

D.

1. Three added to the product of –4 and a number x is less than 5 added to

the product of –3 and the same number x. Which of the following is a graph

of the solution set of x?

p. 359

Graphing Inequalities

Graphing on the Coordinate Plane

The coordinate plane is formed by two intersecting lines: an x-axis and a y-axis

The x-axis is horizontal and the y-axis is vertical

The point at which the x-axis and y-axis meet is called the origin

p. 364

Graphing on the Coordinate Plane

Each point in the coordinate plane can be named using an ordered pair

The first number is the distance from the origin along the x-axis

The second number is the distance from the origin along the y-axis

P is at (–1, –3) and M is at (2, 3)

p. 364

A. (0, –3)

B. (–3, 0)

C. (0, 3)

D. (3, 0)

2. What is the ordered pair for Point C?

p. 365

Graphing on the Coordinate Plane

Slope

When the graph of an equation is a straight line, the equation is a linear

equation

The slope of the line is measured by dividing the difference in y-

coordinates by the difference in x-coordinates, or finding the “rise over

run”

The slope formula can be found on the GED Formula Sheet, but you

should plan on memorizing it:

The slope of the line with points (2, 2) and (1, –1) is:

p. 368

2 1

2 1

y ym

x x

1 2 33

1 2 1m

Slope

Parallel lines always have the same slope

Parallel lines never intersect!

Perpendicular lines have negative reciprocal slopes

Perpendicular lines intersect at a 90 angle

Perpendicular Lines

x

y

slope = 2

slope = 1

2

Parallel Lines

x

y

slope = 2 slope = 2

A. 0.055 calls per hour

B. 17.25 calls per hour

C. 18.25 calls per hour

D. 69 calls per hour

Slope

3. A telemarketing company tracks employee efficiency. Use the graph to

find the rate of calls made per hour of work.

p. 375

Slope and Equations

You can use the slope and a point on the line to find the equation of a line in

two different forms:

Slope-intercept form: y = mx + b

m = slope

b = y-intercept

Point-slope form: y – y1 = m (x – x1)

m = slope

x1 is the x-coordinate of a point

y1 is the y-coordinate of a point

p. 370

x = x-coordinate of a solution

y = y-coordinate of a solution

A. y = –2x + 4

B. y = –4x + 2

C. y = 2x – 4

D. y = 4x – 2

Slope and Equations

4. Find the equation of the line that passes through (1, –2) and has a slope

of –4.

p. 371

Systems of Equations

A system of equations is a set of two or more linear equations

6x + 3y = 12

5x + y = 7

The solution to a system of two equations is the point at which the lines intersect

A system of equations can be solved in multiple ways:

• Graphing

• Substitution

• Combination

p. 376

A. (2, –1)

B. (3, –2)

C. (–2, 1)

D. (–1, 2)

Systems of Equations

5. Where does the line with the equation x – 2y = 4 intersect with the line

with the equation 6y + 5x = 4?

p. 377

Systems of Equations

A. (2, –1)

B. (3, –2)

C. (–2, 1)

D. (–1, 2)

5. Where does the line with the equation x – 2y = 4 intersect with the line

with the equation 6y + 5x = 4?

p. 377

A. (0, 3)

B. (1, 3)

C. (2, 3)

D. (2, 5)

Mixed Practice

6. Line L passes through point (1, 0) and has a slope of 3. Which of the

following points also lies on line L?

p. 369

A. 80 hours per square yard

B. 0.0375 hour per square yard

C. 0.01625 hour per square yard

D. 0.0125 hour per square yard

Mixed Practice

7. On three carpet installation jobs, a contractor records the values below with the

goal of analyzing whether her rates are high enough given the time it takes her to

install carpet. It takes her 1 hour to collect tools and drive to a worksite. The x-axis

shows square yards of carpet, and the y-axis shows hours spent. What is her rate of

time per square yard?

p. 375

Quadrilaterals

Two-dimensional (2D) figures are shapes made up of line segments, such as

rectangles, triangles, and circles

p. 388

rectangle square parallelogram

rhombustrapezoid

Triangles

A triangle has three sides, three angles, and three vertices

Triangles can be classified by side length or angle measure

p. 390

Triangles

A right triangle has one right angle

The side directly across from the right angle is the longest side of the right

triangle and is called the hypotenuse

The two sides adjacent to the right angle are called the legs of the triangle

p. 392

When a and b are used to represent the lengths of the legs and c is used to represent the length of

the hypotenuse, the side lengths of a right triangle can be represented by the equation a2 + b2 = c2

This equation is known as the Pythagorean theorem

8. One side of a right triangle is 3 meters, and the hypotenuse of the triangle

is 6 meters. Find the length of the remaining side to the nearest tenth unit.

You MAY use a calculator.

Triangles

p. 393

Perimeter and Area

Perimeter is the distance around a figure

Perimeter is measured in units

To find perimeter, simply add the lengths of the

sides

Area is the measure of the space inside a flat figure

Area is measured in square units

Certain shapes have formulas to make it easier

to find the perimeter and area of the shape

p. 394

Perimeter and Area

For certain 2D figures, you can apply a formula to find the perimeter

These formulas are provided, but try to memorize them:

p. 394

Perimeter and Area

For certain 2D figures, you can apply a formula to find the area

These formulas are provided, but try to memorize them:

p. 394

Circles

A circle is a closed set of points that are all the same distance from a single

point: the center of the circle

Any line that goes from the center of the circle to a point on the circle is

called the radius

Any line that goes from one point on the circle to another point on the circle

while passing directly through the center is called the diameter

When two radii are combined, they create the diameter

p. 396

Circles

The perimeter of a circle is called the circumference

The area of a circle is still called area

p. 396

circumference = π × diameter C = πd = π (2r)

area = π x (radius)2, or A = πr2

C = circumference d = diameter r = radius A = area

p. 397

9. Find the circumference and area of the circle. Round your answer to the

nearest tenth.

Circles

Three-Dimensional Figures

Three-dimensional (3D) figures are shapes that have height, width and depth.

The measure of space inside the shape is called the volume, and the measure

of the total area of the faces is called the surface area

p. 398

rectangular prism cube cylinderright prism*

sphere pyramid cone

Volume and Surface Area

For certain 3D figures, you can apply a formula to find the surface area and

volume

These formulas are provided, but consider memorizing them:

p. 398

10. A box has two identical rectangular bases. Find the surface area and

volume.

p. 398

Prisms

11. A pyramid is a three-dimensional object with four triangle faces that

connect to the same vertex. Find the surface area of this pyramid.

p. 400

Pyramids

12. A pyramid is a three-dimensional object with four triangle faces that

connect to the same vertex. Find the volume of this pyramid.

p. 400

Pyramids

13. A cylinder has two circular bases connected by a curved surface. Find the

surface area and volume of this cylinder.

p. 399

Cylinders

14. A cone is similar to a cylinder. The curved side of a cone slants inward so

that it meets at a point, or vertex. Find the surface area of the first cone, and

the volume of the second cone below.

p. 405

Cones

15. A sphere is a round solid figure where every point on the surface is the

same distance from the center. Find the surface area and volume.

p. 404

Spheres

A. 16.0

B. 13.7

C. 12.8

D. 12.1

Mixed Practice

16. Jan has built a rectangular frame out of wood to use for the bottom of a

platform. He wants to add a diagonal brace as shown in the drawing below.

What will the length of the brace be to the nearest tenth of a foot?

p. 393

A. 5

B. 10

C. 16

D. 20

Mixed Practice

17. A rectangular box with a volume of 80 cubic feet has the length and

width shown in the drawing. What is the height of the box?

p. 399

A. 6

B. 13

C. 19

D. 113

Mixed Practice

18. On the target below, the 5- and 10-point bands are each 2 inches wide,

and the 25-point inner circle has a diameter of 2 inches. To the nearest inch,

what is the outer circumference of the 10-point band?

p. 397

A. ( –3, –2)

B. ( –3, 2)

C. ( –2, –3)

D. (3, –2)

Mixed Practice

19. Two of the corners of a triangle are located at (3, –3) and (2, 3). What is

the location of the third corner as shown in the diagram below?

p. 365

A. 21

B. 48

C. 67

D. 268

Mixed Practice

20. The height of a cone is half the diameter of its base. If the cone’s height

is 4 inches, what is the cone’s volume to the nearest cubic inch?

p. 401

A. 19

B. 36

C. 38

D. 57

Mixed Practice

21. If workers lay a tile border around the edge of the fountain shown in the

diagram, how many feet long will the border be to the nearest foot?

p. 397

A. y = –x – 1

B. y = 3x + 3

C. y = –3x + 9

D. y = x – 5

Mixed Practice

22. Which of the following is an equation for the line that passes through

(–1, 0) and (2, –3)?

p. 371

A. 5

B. 6

C. 25

D. 50

Mixed Practice

23. All the edges of a metal box are of equal length. If the surface area is

150 square inches, what is the length, in inches, of each edge of the box?

p. 403

A. 108

B. 162

C. 324

D. 432

Mixed Practice

24. Martin is building a rectangular patio centered on one side of his yard.

The rest of his yard, shown in the diagram, is planted with grass. If the

measurements in the diagram are in feet, what is the square footage of the

grass portion of Martin’s yard?

p. 395

Learning Objectives

Now that you have completed this lesson, you should be able to:

• Graph linear equations

• Graph linear inequalities

• Connect and interpret graphs and functions

• Connect coordinates, lines, and equations

• Calculate dimensions, perimeter, circumference, and area of two-dimensional figures

• Calculate dimensions, surface area, and volume of three-dimensional figures

Preparing for Test Day

Your preparation doesn’t end here! In the days leading up to Test Day:

1. Focus on your strengths. You won’t be able to master quadratic equations in two days, but you

might be able to learn how to use all of the geometric formulas.

2. Keep a steady sleep schedule. It’s better to be well-rested on Test Day than to have crammed

more information into your brain at the last minute.

3. Pack your bag the night before and visit the testing center ahead of time, if possible. You must

bring an acceptable photo ID, and you should bring your TI-30XS Multiview Scientific calculator.

Consider dressing in layers. Visit the GED Test Taker portal (www.ged.com) Test Tips What

To Bring for more information.

Homework

Don’t stop now! Practice is important.

To ensure you understand today’s lessons, do the following for homework:

• Quiz 1 – Geometry Practice Questions

p. 410-411, #1-10

• Quiz 2 – Geometry Practice Questions

p. 411-412, #11-19

• Quiz 3 – Equations, Inequalities, and Functions Practice Questions

p. 384-387, #3, 4, 6, 7, 13, 15, 16, 20-23, 26, 32

Answer Key

1. D

2. A

3. C

4. B

5. A

6. C

7. D

8. 5.2

9. 25.1, 50.2

10. 94, 60

11. 39

12. 50

13. 80π, 75π

14. 96π, 12π

15. 40π, 1333.33π

16. D

17. A

18. C

19. A

20. C

21. C

22. A

23. A

24. C

A.

B.

C.

D.

1. Three added to the product of –4 and a number x is less than 5 added to

the product of –3 and the same number x. Which of the following is a graph

of the solution set of x?

3 – 4x < 5 – 3x

3 < 5 + x

–2 < x

x > –2

p. 359

Graphing Inequalities

A. (0, –3)

B. (–3, 0)

C. (0, 3)

D. (3, 0)

2. What is the ordered pair for Point C?

0 units left or right x-coordinate = 0

3 units down y-coordinate = –3

ordered pair: (0, –3)

p. 365

Graphing on the Coordinate Plane

A. 0.055 calls per hour

B. 17.25 calls per hour

C. 18.25 calls per hour

D. 69 calls per hour

Slope

3. A telemarketing company tracks employee efficiency. Use the graph to

find the rate of calls made per hour of work.

Point 1: (0, 0)

Point 2: (4, 73)

m =

m =

m =

m = 18.25

y2 – y1

x2 – x1

73 – 0

4 – 0

73

4

p. 375

A. y = –2x + 4

B. y = –4x + 2

C. y = 2x – 4

D. y = 4x – 2

Slope and Equations

4. Find the equation of the line that passes through (1, –2) and has a slope

of –4.

Slope-Intercept Form:

y = mx + b

–2 = (–4)(1) + b

–2 = –4 + b

b = 2

y = –4x + 2

Point-Slope Form:

y – y1 = m (x + x1)

y – (–2) = (–4)(x + 1)

y + 2 = –4x – 4

y = –4x + 2

p. 371

A. (2, –1)

B. (3, –2)

C. (–2, 1)

D. (–1, 2)

Systems of Equations

5. Where does the line with the equation x – 2y = 4 intersect with the line

with the equation 6y + 5x = 4?

Step 1: Multiply the first equation by 3 to

make the y-coefficients opposites

3 (x – 2y = 4)

6y + 5x = 4

Step 2: Combine the equations by adding the

first equation to the second equation

3x – 6y = 12

+ 5x + 6y = 4

8x + 0y = 16

x = 2

Step 3: Substitute x into either

equation to solve for y

x – 2y = 4

2 – 2y = 4

– 2y = 2

y = –1

Combination

p. 377

6y + 5x = 4

6y + 5 (2) = 4

6y + 10 = 4

6y = –6

y = –1

OR

3x – 6y = 12

5x + 6y = 4

Systems of Equations

A. (2, –1)

B. (3, –2)

C. (–2, 1)

D. (–1, 2)

5. Where does the line with the equation x – 2y = 4 intersect with the line

with the equation 6y + 5x = 4?

Step 1: Get x by itself in either equation

x – 2y = 4

x = 2y + 4

Step 2: Substitute the equivalent

expression into the other equation to solve

for y

6y + 5x = 4

6y + 5 (2y + 4) = 4

6y + 10y + 20 = 4

16y = –16

y = –1

Step 3: Substitute y into either

equation to solve for x

x – 2y = 4

x – 2(–1) = 4

x – (–2) = 4

x + 2 = 4

x = 2

Substitution

p. 377

6y + 5x = 4

6 (–1) + 5x = 4

–6 + 5x = 4

5x = 10

x = 2

OR

A. (0, 3)

B. (1, 3)

C. (2, 3)

D. (2, 5)

Mixed Practice

6. Line L passes through point (1, 0) and has a slope of 3. Which of the

following points also lies on line L?

Line L: y = mx + b

slope = 3

y-intercept = b

point = (1, 0)

y = mx + b

0 = 3 (1) + b

b = –3

Line L: y = 3x – 3

p. 369

Plug each choice into y = 3x –3

Choice A: 3 = 3(0) –3

3 ≠ –3

Choice B: 3 = 3 (1) –3

3 ≠ 0

Choice C: 3 = 3(2) –3

3 = 3 ✓

A. 80 hours per square yard

B. 0.0375 hour per square yard

C. 0.01625 hour per square yard

D. 0.0125 hour per square yard

Mixed Practice

7. On three carpet installation jobs, a contractor records the values below with the

goal of analyzing whether her rates are high enough given the time it takes her to

install carpet. It takes her 1 hour to collect tools and drive to a worksite. The x-axis

shows square yards of carpet, and the y-axis shows hours spent. What is her rate of

time per square yard?

time per square yard = slope

p. 375

hours 2.25 1.5

square yard 100 40

hours 0.75

square yard 60

hour0.0125

square yard

8. One side of a right triangle is 3 meters, and the hypotenuse of the triangle

is 6 meters. Find the length of the remaining side to the nearest tenth unit.

You MAY use a calculator.

leg a = a meters

leg b = 3 meters

hypotenuse c = 6 meters

To find a, take the square root of 27.

a = 5.19615244…

a ≈ 5.2 square meters

a2 + b2 = c2

a2 + 32 = 62

a2 + 9 = 36

a2 = 27

Triangles

p. 393

p. 397

9. Find the circumference and area of the circle. Round your answer to the

nearest tenth.

radius = 4 meters

diameter = 8 meters

Circumference = π × diameter, or C = πd

C = π × 8

C = 8 × 3.14

C = 25.12

C = 25.1 meters

Area = π × (radius)2, or A = πr2

A = π × 42

A = π × 16

A = 16 × 3.14

A = 50.24

A = 50.2 square meters

Circles

10. A box has two identical rectangular bases. Find the surface area and

volume.

Surface area = combined area of all six sides = 2lw + 2lh + 2wh

Volume = length × width × height = lwh

p. 398

SA = 2 (4) (5) + 2 (4) (3) + 2 (5) (3)

= 40 + 24 + 30

= 94 square feet

V = lwh

= (4) (5) (3)

= 60 cubic feet

Prisms

11. A pyramid is a three-dimensional object with four triangle faces that

connect to the same vertex. Find the surface area of this pyramid.

p. 400

p = perimeter of square base

= 3 + 3 + 3 + 3 = 12

s = 5 inches

B = area of square base

B = 32 = 9 square inches

SA = 6 (5) + 9

SA = 30 + 9

SA = 39 square inches

1SA

2ps B

1SA (12)(5) 9

2

Surface area = × perimeter of base × slant height + area of base = ps + B 1

2

1

2

Pyramids

12. A pyramid is a three-dimensional object with four triangle faces that

connect to the same vertex. Find the volume of this pyramid.

p. 400

1Volume

3Bh

1V (5 5) 6

3

50 cubic cm

Volume = × area of base × height perpendicular to base = Bh 1

3

1

3

Pyramids

13. A cylinder has two circular bases connected by a curved surface. Find the

surface area and volume of this cylinder.

Surface area = 2π × radius × height + 2π × (radius)2 = 2πrh + 2πr2

Volume = π × (radius)2 × height = πr2h

p. 399

SA = 2πrh + 2πr2

= 2π (5) (3) + 2π (5)2

= 2π (15) + 2π (25)

= 30π + 50π = 80π square units

V = πr2h

= π (5)2 × (3)

= 75π cubic units

Cylinders

14. A cone is similar to a cylinder. The curved side of a cone slants inward so

that it meets at a point, or vertex. Find the surface area of the first cone, and

the volume of the second cone below.

p. 405

V = πr2h

V = π (22) (9)

V = 12π cubic inches

1

3

1

3

SA = πrs + πr2

= π (8) (4) + π (8)2

= 32π + 64π

= 96π square units

Surface area = π × radius × slant + π × (radius) 2 = πrs + πr2

Volume = × π × (radius)2 × height perpendicular to base = πr2h1

3

1

3

Cones

15. A sphere is a round solid figure where every point on the surface is the

same distance from the center. Find the surface area and volume.

Surface area = 4π × (radius)2 = 4πr2

Volume = × π × (radius)3 = πr3

p. 404

4

3

4

3

V = πr3

= π (103)

≈ 1333.33π cubic units

4

3

4

3

SA = 4πr2

= 4π (10)2

= 400π square units

Spheres

A. 16.0

B. 13.7

C. 12.8

D. 12.1

Mixed Practice

16. Jan has built a rectangular frame out of wood to use for the bottom of a

platform. He wants to add a diagonal brace as shown in the drawing below.

What will the length of the brace be to the nearest tenth of a foot?

diagonal brace = hypotenuse

a2 + b2 = c2

112 + 52 = c2

121 + 25 = c2

146 = c2

12.1 feet ≈ c

p. 393

c

A. 5

B. 10

C. 16

D. 20

Mixed Practice

17. A rectangular box with a volume of 80 cubic feet has the length and

width shown in the drawing. What is the height of the box?

V = lwh

volume = 80

length = 4

width = 4

height = h

80 = 4 × 4 × h

80 = 16h

5 feet = h

p. 399

A. 6

B. 13

C. 19

D. 113

Mixed Practice

18. On the target below, the 5- and 10-point bands are each 2 inches wide,

and the 25-point inner circle has a diameter of 2 inches. To the nearest inch,

what is the outer circumference of the 10-point band?

diameter = 2 + 2 + 2 = 6

C = πd

C ≈ 3.14 × 6

C ≈ 18.84

C ≈ 19 inches

p. 397

22 2

A. ( –3, –2)

B. ( –3, 2)

C. ( –2, –3)

D. (3, –2)

Mixed Practice

19. Two of the corners of a triangle are located at (3, –3) and (2, 3). What is

the location of the third corner as shown in the diagram below?

The point is three units to the left of the origin,

so the x-coordinate is –3

The point is two units down from the origin, so the

y-coordinate is –2

(x, y) = (–3, –2)

p. 365

(3, –3)

(2, 3)

(x, y)

volume = π × (radius)2 × height

volume = π × (4)2 × 4

volume = π × 16 × 4

volume ≈ × 3.14 × 64

volume ≈ 67.2 in3

A. 21

B. 48

C. 67

D. 268

Mixed Practice

20. The height of a cone is half the diameter of its base. If the cone’s height

is 4 inches, what is the cone’s volume to the nearest cubic inch?

height = 4

the height is half the diameter of its base:

4 = (diameter)

d = 8 inches

radius = half the diameter = (8) = 4 inches

p. 401

1

2

1

2

1

3

1

3

1

31

3

A. 19

B. 36

C. 38

D. 57

Mixed Practice

21. If workers lay a tile border around the edge of the fountain shown in the

diagram, how many feet long will the border be to the nearest foot?

Border of the fountain =

circumference of the circle

C = πd

C = π x 12

C ≈ 3.14 x 12

C ≈ 37.68

C ≈ 38 feet

p. 397

A. y = –x – 1

B. y = 3x + 3

C. y = –3x + 9

D. y = x – 5

Mixed Practice

22. Which of the following is an equation for the line that passes through

(–1, 0) and (2, –3)?

slope-intercept form for a line: y = mx + b

points: (–1, 0) and (2, –3)

p. 371

y-intercept = b

Plug in m and one point:

y = mx + b

0 = –1(–1) + b

0 = 1 + b

–1 = b

final equation: y = –x – 1

2 1

2 1

slopey y

mx x

3 0

2 ( 1)m

3

3

1

A. 5

B. 6

C. 25

D. 50

Mixed Practice

23. All the edges of a metal box are of equal length. If the surface area is

150 square inches, what is the length, in inches, of each edge of the box?

All the edges are of equal length = square sides

Surface area of box = 6 (area of one square)

SA = 6s2

150 = 6s2

25 = s2

5 inches = s

p. 403

A. 108

B. 162

C. 324

D. 432

Mixed Practice

24. Martin is building a rectangular patio centered on one side of his yard.

The rest of his yard, shown in the diagram, is planted with grass. If the

measurements in the diagram are in feet, what is the square footage of the

grass portion of Martin’s yard?

Grass portion is everything in the yard except the patio

Grass = yard – patio

Area of the yard = 18 x 24 = 432

Area of the patio = 9 x 12 = 108

Grass = 432 – 108 = 324 square feet

p. 395