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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