Download - Triumph Learning

Coach is the leader in standards-based, state-customized instruction for grades K–12 in English
language arts, mathematics, science, and social studies. Our student texts deliver everything you
need to meet your state standards and prepare your class for grade-level success!
Coach lessons have just what you’re looking for:
✔✔ Easy-to-follow, predictable lesson plans
Florida Coach, Standards-Based Instruction, Geometry
Standards-Based Curriculum Support!
✔✔ Focused instruction with modeled examples
✔✔ Guided practice with hints and support
✔✔ Higher-level thinking activities
PLUS Chapter Reviews that target assessed skills
Used by more students in the U.S. than any other state-customized series, Coach books are
proven effective. Triumph Learning has been a trusted name in educational publishing for more than
40 years, and we continue to work with teachers and administrators to keep our books up to date—
improving test scores and maximizing student learning.
Please visit our Web site for detailed product descriptions of all our instructional materials, including
sample pages and more.
www.triumphlearning.com
Phone: (800) 221-9372 • Fax: (866) 805-5723 • E-mail: [email protected]
188FL_Geo_HS_SE_Cvr.indd 1
188FL
This book is printed on paper containing
a minimum of 10% post-consumer waste.
Developed in Consultation
with Florida Educators
12/3/09 11:02:04 AM
188FL_Mth_Geo_SE_FM_PDF.indd Page 3
9/26/09
9:34:58 AM u-s008
/Volumes/109/TL00110/work%0/indd%0/SE/FM
Table of Contents
Benchmarks
Letter to the Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Test-Taking Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Florida Mathematics Standards Correlation Chart . . . . . . . . . . . . . . . . . 8
Chapter 1
Mathematical Reasoning and Logic . . . . . . . . . . . . . . . . 13
Lesson 1
Introduction to Euclidean Geometry . . . . . . . . . 14
MA.912.G.8.1
Lesson 2
Converses, Inverses, and
Contrapositives . . . . . . . . . . . . . . . . . . . . . . . . . 18
MA.912.D.6.2
Geometric Proofs . . . . . . . . . . . . . . . . . . . . . . . 24
MA.912.D.6.4, MA.912.G.8.2,
Lesson 3
MA.912.G.8.4, MA.912.G.8.5
Lesson 4
Logical Equivalence and Validity . . . . . . . . . . . 29
MA.912.D.6.3, MA.912.D.6.4,
MA.912.G.8.3
Chapter 1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Chapter 2
Points, Lines, Angles, and Planes . . . . . . . . . . . . . . . . . . 39
Lesson 5
Distance and Midpoint . . . . . . . . . . . . . . . . . . . 40
MA.912.G.1.1
Lesson 6
Constructing Congruent Segments
and Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
MA.912.G.1.2, MA.912.G.8.6
Constructing Bisectors and Parallel
and Perpendicular Lines . . . . . . . . . . . . . . . . . . 50
MA.912.G.1.2, MA.912.G.4.2,
Lesson 7
MA.912.G.8.6
Lesson 8
Parallel Lines and Transversals. . . . . . . . . . . . . 56
MA.912.G.1.3
Chapter 2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Chapter 3
Polygons and Quadrilaterals. . . . . . . . . . . . . . . . . . . . . . . 69
Lesson 9
Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
MA.912.G.2.1, MA.912.G.2.2
Lesson 10 Solving Problems with Congruent
and Similar Polygons . . . . . . . . . . . . . . . . . . . . 75
MA.912.G.2.3, MA.912.G.8.2
Lesson 11 Transformations, Congruence, Similarity,
and Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 81
MA.912.G.2.4
Lesson 12 Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
MA.912.G.2.4
Lesson 13 Perimeter and Area of Polygons . . . . . . . . . . . . 93
MA.912.G.2.5
Duplicating any part of this book is prohibited by law.
3
P DF P ass
188FL_Mth_Geo_SE_FM_PDF.indd Page 4
9/26/09
9:34:59 AM u-s008
/Volumes/109/TL00110/work%0/indd%0/SE/FM
Lesson 14 Changes in the Dimensions of
Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
MA.912.G.2.7
Lesson 15 Special Quadrilaterals . . . . . . . . . . . . . . . . . . . 102
MA.912.G.3.1, MA.912.G.3.2
Lesson 16 Proving Properties of Quadrilaterals . . . . . . . . 109
MA.912.G.3.3
Lesson 17 Proving Quadrilateral Theorems . . . . . . . . . . . 115
MA.912.G.3.4
Chapter 3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Chapter 4
Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Lesson 18 Classifying and Constructing
Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
MA.912.G.4.1, MA.912.G.8.6
Lesson 19 Points of Concurrency in Triangles . . . . . . . . . 132
MA.912.G.4.2, MA.912.G.8.6
Lesson 20 Constructing Congruent Triangles . . . . . . . . . 139
MA.912.G.4.3, MA.912.G.8.6
Lesson 21 Solving Problems with Congruent
and Similar Triangles. . . . . . . . . . . . . . . . . . . . 144
MA.912.G.4.4, MA.912.G.4.5,
MA.912.G.8.2
Lesson 22 Proving Congruence and Similarity . . . . . . . . 149
MA.912.G.4.5, MA.912.G.4.6
Lesson 23 Triangle Inequality Theorems . . . . . . . . . . . . . 156
MA.912.G.4.7
Lesson 24 Right Triangle Theorems . . . . . . . . . . . . . . . . . 161
MA.912.G.5.1, MA.912.G.5.2
Lesson 25 Pythagorean Theorem . . . . . . . . . . . . . . . . . . 165
MA.912.G.5.1, MA.912.G.5.4
Lesson 26 Trigonometric Ratios and Special
Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . 169
MA.912.G.5.3, MA.912.G.5.4,
MA.912.T.2.1
Chapter 4 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Chapter 5
Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Lesson 27 Parts of a Circle . . . . . . . . . . . . . . . . . . . . . . . 182
MA.912.G.6.2
Lesson 28 Measures of Arcs and Related
Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
MA.912.G.6.4
Lesson 29 Solving Real-World Problems
with Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
MA.912.G.6.5, MA.912.G.8.2
Lesson 30 Equations of Circles . . . . . . . . . . . . . . . . . . . . 197
MA.912.G.6.6, MA.912.G.6.7
Chapter 5 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
4
Duplicating any part of this book is prohibited by law.
P DF P ass
188FL_Mth_Geo_SE_FM_PDF.indd Page 5
9/26/09
9:34:59 AM u-s008
/Volumes/109/TL00110/work%0/indd%0/SE/FM
Table of Contents
Chapter 6
Polyhedra and Other Solids . . . . . . . . . . . . . . . . . . . . . . 207
Lesson 31 Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
MA.912.G.7.1, MA.912.G.7.2
Lesson 32 Nets of Polyhedra . . . . . . . . . . . . . . . . . . . . . . 215
MA.912.G.7.1
Lesson 33 Spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
MA.912.G.7.4
Lesson 34 Surface Area and Volume . . . . . . . . . . . . . . . . 226
MA.912.G.7.5
Lesson 35 Changes in the Dimensions of Solids . . . . . . . 232
MA.912.G.7.7
Lesson 36 Congruent and Similar Solids . . . . . . . . . . . . . 237
MA.912.G.7.6
Chapter 6 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
Duplicating any part of this book is prohibited by law.
5
P DF P ass
188FL_Mth_Geo_SE_Ch04_PDF.indd Page Sec1:132
19
9/26/09
1:27:30 PM u-s008
/Volumes/109/TL00110/work%0/indd%0/SE/Chapter 04
Points of Concurrency
in Triangles
MA.912.G.4.2, MA.912.G.8.6
If three or more lines are concurrent, then they intersect at the same point, which is called the
point of concurrency. Knowing about four points of concurrency in a triangle can help you solve
many different types of problems in mathematics and in the real world.
The incenter is the point at which the angle bisectors of a triangle intersect. An angle bisector is
a line or ray that divides an angle in half. The incenter, as its name suggests, is also the center of
the circle that can be inscribed in the triangle.
angle bisector
incenter
angle bisector
angle bisector
The orthocenter is the point at which the three altitudes of a triangle intersect. An altitude is the
shortest distance between a vertex of a triangle and the opposite side (even if the opposite side
needs to be extended to find that shortest distance). The shortest distance is always perpendicular
to the opposite side.
altitude
altitude
orthocenter
altitude
132
Duplicating any part of this book is prohibited by law.
P DF P ass
188FL_Mth_Geo_SE_Ch04_PDF.indd Page Sec1:133
9/26/09
1:27:31 PM u-s008
/Volumes/109/TL00110/work%0/indd%0/SE/Chapter 04
Lesson 19: Points of Concurrency in Triangles
The circumcenter is the point at which the three perpendicular bisectors of the sides of a
triangle intersect. A perpendicular bisector is a line that is perpendicular to a line segment and
passes through the midpoint of the line segment. The circumcenter is also the center of the
circle that could be circumscribed around the triangle such that all three vertices of the triangle
are on the circle. Also, the circumcenter is equidistant (the same distance) from each of the
triangle’s vertices.
perpendicular
bisector
perpendicular
bisector
circumcenter
perpendicular
bisector
The centroid is the point at which the medians of a triangle intersect. The median of a triangle is
a line segment drawn from a vertex of a triangle to the midpoint of the opposite side. The centroid
divides each median into two segments, and the longer segment is twice as long as the shorter
segment. The centroid is also the center of gravity of the triangle.
midpoint
midpoint
centroid
midpoint
You can construct each of these points of concurrency, using tools such as a compass.
Duplicating any part of this book is prohibited by law.
133
P DF P ass
188FL_Mth_Geo_SE_Ch04_PDF.indd Page Sec1:134
9/26/09
1:27:31 PM u-s008
/Volumes/109/TL00110/work%0/indd%0/SE/Chapter 04
EXAMPLE 1
Construct the incenter of XYZ shown below.
X
Z
STRATEGY
STEP 1
Y
Use a compass to draw an angle bisector for each vertex of the angle. Then
find the point at which those angle bisectors intersect.
Construct an angle bisector for vertex X.
Draw an arc from vertex X so that the arc crosses both sides of the angle. Plot
points where the arc crosses each side.
Draw an arc from each of the points you plotted into the triangle’s interior, using
the same compass opening each time. Plot a point where the two arcs meet.
Finally, draw a line segment from vertex X through the intersection of the two
arcs to the opposite side. This is an angle bisector.
X
X
X
Y
Z
Z
Y
Z
Y
STEP 2
Using the steps described above, construct angle bisectors for vertices Y and Z.
STEP 3
Label the point where the angle bisectors intersect as the incenter.
Note: You can inscribe a circle in the triangle, using the incenter as its center.
X
X
incenter
Z
SOLUTION
134
incenter
Y
Z
Y
The construction in Step 3 shows the incenter of XYZ.
Duplicating any part of this book is prohibited by law.
P DF P ass
188FL_Mth_Geo_SE_Ch04_PDF.indd Page Sec1:135
9/26/09
1:27:31 PM u-s008
/Volumes/109/TL00110/work%0/indd%0/SE/Chapter 04
Lesson 19: Points of Concurrency in Triangles
EXAMPLE 2
Evansville
A county wants to build a new public library.
They want to build it at a location that is equidistant
from the centers of three towns—Derry, Evansville,
and Franklinton—shown on the map below. Which
point of concurrency could you draw on this map to
show the ideal location for the library?
STRATEGY
STEP 1
Franklinton
Derry
Consider which point of concurrency is the same distance from all the vertices
of a triangle. Construct this point.
Determine which point of concurrency is equidistant from the vertices.
The circumcenter of a triangle is the center of a circle which has all three
vertices of the triangle on the circle.
The distance from each vertex to the circumcenter is equal to a radius of
the circle.
Each radius is the same length, so each vertex will be equidistant from
the circumcenter.
STEP 2
Construct the circumcenter of the triangle.
Construct the perpendicular bisector of the segment joining Derry and Evansville.
Evansville
Franklinton
Derry
Repeat this process for the other two sides.
radii
Evansville
radius
Franklinton
Derry
SOLUTION
circumcenter
The circumcenter of the triangle is the ideal location for the library.
Duplicating any part of this book is prohibited by law.
135
P DF P ass
188FL_Mth_Geo_SE_Ch04_PDF.indd Page Sec1:136
9/26/09
1:27:32 PM u-s008
/Volumes/109/TL00110/work%0/indd%0/SE/Chapter 04
EXAMPLE 3
___
Point T is the centroid of MNP. If TR 6 cm, what is the length of MT?
N
Q
M
STRATEGY
STEP 1
T
R
P
S
Use what you know about medians and centroids.
What do you know about medians and centroids?
The centroid of a triangle divides each median into two segments. The longer
segment is twice the length of the shorter segment.
___
STEP 2
Find the length of MT.
____
Centroid T divides the median MR into two segments.
___
___
The shorter segment, TR, is 6 cm long, so the longer segment, MT, will be twice
that length.
___
MT 6 2 12 cm
SOLUTION
____
The length of MT is 12 cm.
COACHED EXAMPLE
___
___ ___
___
___
___
In the diagram below, AH BC, BJ AC, and CK AB. What point of concurrency is point M ?
B
K
H
M
A
J
C
THINKING IT THROUGH
The line segment from the vertex of a triangle that is perpendicular to the opposite side is a(n)
of the triangle.
intersect at the point of concurrency called the
The three
___ ___
Since point M shows the intersection of
.
___
AH, BJ, and CK, it is the
of ABC.
136
Duplicating any part of this book is prohibited by law.
P DF P ass
188FL_Mth_Geo_SE_Ch04_PDF.indd Page Sec1:137
9/26/09
1:27:32 PM u-s008
/Volumes/109/TL00110/work%0/indd%0/SE/Chapter 04
Lesson 19: Points of Concurrency in Triangles
Lesson Practice
Choose the correct answer.
1.
In the diagram below, lines DE, FG, and
HJ are perpendicular bisectors of the sides
of ABC.
3.
____
If MD 20 units, what is the length of DA?
A. 10 units
B. 20 units
B
C. 30 units
F
D
D. 40 units
H
K
A
4.
G
J
Point Z is the orthocenter of TUV.
C
E
U
X
Which describes point K ?
Y
Z
A. centroid
B. circumcenter
C. incenter
T
D. orthocenter
Point D is the centroid of KLM below.
K
D
M
2.
V
____
Which term accurately describes TY ?
Use this diagram for questions 2 and 3.
C
W
A
altitude
B
angle bisector
C
median of triangle
D
perpendicular bisector
A
B
L
_____
If KC 22 units, what is the length of CM ?
A. 11 units
B. 20 units
C. 22 units
D. 44 units
Duplicating any part of this book is prohibited by law.
137
P DF P ass
188FL_Mth_Geo_SE_Ch04_PDF.indd Page Sec1:138
9/26/09
1:27:33 PM u-s008
Use the map below for questions 5 and 6.
Briarville
Lo
vie
Ma
in
Roa
d
ng
Avon
5.
Peach Road
w
6.
/Volumes/109/TL00110/work%0/indd%0/SE/Chapter 04
A school is equidistant from each of the
three towns shown on the map. Which
could be used to find the school’s location?
A. centroid
Ro
ad
B. circumcenter
Claytown
D. orthocenter
C. incenter
A museum is equidistant from each of the
three roads shown on the map. Which could
be used to find the museum’s location?
A. centroid
B. circumcenter
C. incenter
D. orthocenter
138
Duplicating any part of this book is prohibited by law.
P DF P ass