^ chapter ^ get ready! · 2020. 9. 10. · ^ chapter ^ get ready! skills handbook, p. 889 skills...
TRANSCRIPT
^ CHAPTER ^
Get Ready!Skills
Handbook,
p. 889
Skills
Handbook,p. 890
Skills
Handbook,p. 890
Skills
Handbook,p. 892
Skills
Handbook,p. 894
^ Squaring NumbersSimplify.
1. 3^ 2. 4' 3. 11'
^ Simplifying ExpressionsSimplify each expression. Use 3.14 for tt.
4.2-7.5 + 2-11 5.
Evaluating ExpressionsEvaluate the following expressions for a = 4 and b
7.a + b
8.a-7
3-b
6. VPTT?
-2.
9. V(7-fl)2 + [2-fo)2
^ Finding Absolute ValueSimplify each absolute value expression.
10. |-8| 11. |2-6|
^ Solving EquationsAlgebra Solve each equation.
13. 2x+7 = 13 14. 5x-12 = 2x + 6
12. 1-5-(-8)1
15. 2(a: + 3) - 1 = 7a;
Looking Ahead Vocabulary16. A child can construct models of buildings by stacking and arranging colored blocks.
What might the term construction mean in geometry?
17. The Mid-Autumn Festival, celebrated in China, falls exactly in the middle ofautumn, according to the Chinese lunar calendar. What would you expect amidpoint to be in geometry?
18. Artists often use long streaks to show rays of light coming from the sun. A ray is alsoa geometric figure. What do you think the properties of a ray are?
19. You and your friend work with each other. In other words, you and your friend areco-workers. What might the term collinear mean in geometry?
C I chapter 1 Tools of Geometry^
CHAPTER
f _ > Tools of Geometry
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1-1
1-2
1-3
1-4
1-5
1-6
1-7
1-8
r PreviewNets and Drawings for Visualizing
Geometry
Points, Lines, and Planes
Measuring Segments
Measuring Angles
Exploring Angle Pairs
Basic Constructions
Midpoint and Distance in the
Coordinate Plane
Perimeter, Circumference, and Area
BIG ideas
VocabularyEnglish/Spanish Vocabulary Audio Online:
English Spanish
angle bisector, p. 37 bisectriz de un anqulo
construction, p. 43 construccion
isometric drawing, p. 5 dibujo isometrico
linear pair, p. 36 par lineal
net, p. 4 plantilla
orthographic drawing, p, 6 dibujo ortoqrMico
perpendicular bisector, p. 44 mediatriz
postulate, p. 13 postulado
1 Visualization
Essential Question How can yourepresent a three-dimensional figure witha two-dimensional drawing?
2 ReasoningEssential Question What are the
building blocks of geometry?
3 Measurement
Essential Question How can you
describe the attributes of a segment or
angle?
DOMAINS
• Congruence
congruent segments, p. 22 segmentos conqruentes
segment bisector, p. 2i bisectriz de un seqmento
supplementary angles, p. 34
vertical angles, p. 34
angulos suplementarlos
anqulos verticales
Virtual Nerd™
tutorials withbuilt-in support
Common Core Performance Task
Solving a RiddleWhile browsing in an antique store, Cameron found a page that came from an oldbook of riddles.
What sits in a corner but travels
around the world?
(8m - 18)(7m + 3
(21s + 6)
5a + 12 9a - 12
Solve the riddle, today at the latest.Arrange the variables from least to greatest.
Task DescriptionFind the value of each variable and answer the riddle.
Connecting the Task to the Math PracticesAs you complete the task, you'll apply several Standards for MathematicalPractice.
• You'll make sense of the problem as you analyze the information in a complexdiagram. (MP 1)
• You'll attend to precision as you write and solve an equation. (MP 6)
• You'll explain relationships between angles in the diagram. (MP 3)
MATHEMATICAL
PRACTICES
0PqwerGMmet^com I Chapter 1 Tools of Geometry
Nets and Drawings forVisualizing Geometry
Mathematics Fiorlda StandardsPrepares for MAFS.912.G-C0.1.1 Know precisedefinitions of angle, circle, perpendicular line, parallelline, and line segment, based on the undefined notionsof point, line, distance along a line, and distance arounda circular arc.
ivip 3, MP 4,"MP"7
Objective To make nets and drawings of three-dimensional figures
Getting Ready! <>■ X ^
Try to visualizewhat the figuremight look likefrom differentperspectives.
When you shine a flashlight onan object, you can see a shadowon the opposite wall. What shapewould you expect the shadows inthe diagram to hove? Explain yourreasoning.
MATHEMATICALPRACTICES
LessonVocabulary
• net
isometricdrawingorthographicdrawing
In the Solve It, you had to "see" the pro]'ection of one side of an object onto a flatsurface. Visualizing figures is a key skill that you will develop in geometry.
Essential Understanding You can represent a three-dimensional object with atwo-dimensional figure using special drawing techniques.
A net is a two-dimensional diagram that you can fold to form a three-dimensionalfigure. A net shows all of the surfaces of a figure in one view.
How can you seethe 3-D figure?Visualize folding the netat the seams so that theedges join together. Trackthe letter positions byseeing one surface movein relation to another.
Identifying a Solid From a Net
Hie net at the right folds into the cube shown beside it.Which letters will be on the top and front of the cube?
A, C, E, and F all share an edge with Dwhen you fold the net, but only two ofthose sides are visible in the cube shown.
A B C D
A wraps around and joins with D to become the back of the cube.B becomes the left side. F folds back to become the bottom.
E folds down to become the top of the cube. C becomes the front.
Chapter 1 Tools of Geometry
Got It? 1. The net in Problem 1 folds into the cube shown at the right.Which letters will be on the top and right side of the cube?
T;r-i ■
How can you seethe net?
Visualize opening the topand bottom flaps of thebox. Separate one of theside seams. Then unfold
and flatten the box
completely.
J
Packaging designers use nets to design boxes and other containers like the box
in Problem 2.
Problem 2 Drawing a Net From a Solid QS)
Package Design What is a net for the graham cracker boxto the right? Label the net with its dimensions. 20 cm
6 cm
14 cm
cfacHcrs
CraviCfatvJ
14 cm
20 cm
^^1^ Got It? 2. a. What is a net for the figure at the right? Label the net withits dimensions,
b. Reasoning Is there another possible net for the figure inpart (a)? If so, draw it.
10 cm
7 cm
10 cm
4 cm
I
JAn isometric drawing shows a corner view of a three-dimensional
figure. It allows you to see the top, front, and side of the figure. You candraw an isometric drawing on isometric dot paper. The simple drawingof a file cabinet at the right is an isometric drawing.
A net shows a three-dimensional figure as a folded-out flat surface.
An isometric drawing shows a three-dimensional figure using slantedlines to represent depth.
C PowerGeometry.com I Lesson 1-1 Nets and Drawings for Visualizing Geometry
Is there more than
one way to make anisometric drawing?Yes. You can start with
any edge of the structure.Use that edge as areference to draw the
other edges.
Problem 3 Isometric Drawing
What is an isometric drawing of the cube structure at the right?
Step 1
Draw the front edges.
Step 2
Draw the right edges.
Step 3
Draw the back edges.
Got It? 3. What is an isometric drawing of this cube structure?
How can you
determine the
three views?
Rotate the structure in
your head so that youcan "see" each of the
three sides straight on.
An orthographic drawing is another way to represent a three-dimensional figure.An orthographic drawing shows three separate views: a top view, a front view, anda right-side view.
Although an orthographic drawing may take more time to analyze, it providesimique information about the shape of a structure.
Orthographic Drawing
What is the orthographic drawing for the isometric drawing at the right?
Solid lines show -
visible edges. i—
□rTop
Dashed linesshow hiddenedges. -rent Right
Oi:
An isometricdrawing shows thesame three views.
4. What is the orthographic drawing for this^ isometric drawing?
Chapter 1 Tools of Geometry
& Lesson Check
Do you know HOW?
1. What is a net for the figure below? Label the net withits dimensions.
2 in. h/1 in.
5 in.
2. What is an isometric drawing of
the cube structure?
3. What is the orthographic
drawing of the isometric
drawing at the right? Assume
there are no hidden cubes.
MATHEMATICAL
PRACTICESDo you UNDERSTAND?
4. Vocabulary Tell whether each drawing is isomefn'c,orthographic, a net, or none.
a.
7\<n>
b.
Top
V Front
&Right
5. Compare and Contrast What are the differences and
similarities between an isometric drawing and anorthographic drawing? Explain.
Practice and Problem-Solving Exercises
© Practice Match each three-dimensional figure with its net.
6. 7. ZEI>
A.
MATHEMATICAL
PRACTICES
C.
^ See Problem 1.
Draw a net for each figure. Label the net with its dimensions.
9.
2 in.
2 in.4 in.
4^ See Problem 2.
11.
30 mmp i
36 mm
C PowerGeometry.com I Lesson 1-1 Nets and Drawings for Visualizing Geometry
Make an isometric drawing of each cube structure on isometric dot paper.
12. 13. 14.
^ See Problem 3.
For each isometric drawing, make an orthographic drawing. Assume there are ^ See Problem 4.no hidden cubes.
16. 17.
Apply @ 20. Multiple Representations There are eight different nets for the solidshown at the right. Draw as many of them as you can. [Hint: Two netsare the same if you can rotate or flip one to match the other.)
21. a. Open-Ended Make an isometric drawing ofa structure that you
can build using 8 cubes.
b. Make an orthographic drawing of this structure.
22. Think About a Plan Draw a net of the can at the right.
• What shape are the top and bottom of the can?• If you uncurl the body of the can, what shape do you get?
23. History In 1525, German printmaker Albrecht Diirer first used
the word net to describe a printed pattern that folds up into a
three-dimensional shape. Why do you think he chose to use
the word netl
^025) Manufacturing Match the package with its net.
24.
A. B.
N
8 Chapter 1 Tools of Geometry
27. Error Analysis Miquela and Gina drew orthographic drawings for the cubestructure at the right. Who is correct?
Miquela Gina
Top Top
■ront Right "rent□
Right
Make an orthographic drawing for each isometric drawing.
28.. • • 29.' . •
w-* .
30.. *
W •
31. Fort Use the diagram of the fort at the right.a. Make an isometric drawing of the fort.b. Make an orthographic drawing of the fort.
^25) 32. Aerial Photography Another perspective in aerialphotography is the "bird's-eye view," which shows an objectfrom direcdy overhead. What type of drawing that you havestudied in this lesson is a bird's-eye view?
' 33. Writing Photographs of buildings are typically not takenfrom a bird's-eye view. Describe a situation in which youwould want a photo showing a bird's-eye view.
Visualization Think about how each net can be folded to form a cube. What isthe color of the face that will be opposite the red face?
34.
R35. 37.
jT38. Multiple Representations There are 11 different nets for a cube. Four of them are
shown above.
a. Draw the other seven nets.
b. Writing Suppose you want to make 100 cubes for an art project. Which of the11 nets would you use? Explain why.
c j PowerGeometry.com Lesson 1-1 Nets and Drawings for Visualizing Geometry
Challenge 39. The net at the right folds into a cube. Sketch the cube so that its front face isshaded as shown below.
40. Architecture What does the net of the staircase shown look like? Draw the net
{Hint. Visualize stretching the stairs out flat.)
41. A hexomino is a two-dimensional figure formed with she squares. Each square
shares at least one side with another square. The 11 nets of a cuhe that you found
in Exercise 38 are hexominoes. Draw as many of the remaining 24 hexominoes
as you can.
42. Visualization Use the orthographic drawing at the right.
a. Make an isometric drawing of the structure.
b. Make an isometric drawing of the structure from part (a) after it has beenturned on its base 90° counterclockwise.
c. Make an orthographic drawing of the structure from part (b).d. Turn the structure from part (a) 180°. Repeat parts (b) and (c).
Top
Front Right
Short
.Response
Standardized Test Prep
43. How many possible nets does the solid at the right have?
®1 ®2 C03
44. Solve lOfl - 5h = 25 for b.
CT:)h=10fl + 25 C^h=10fl-25 CS:)b = 2a + 5
45. Graph the equation x + 2y= -3. Label the x- andy-intercepts.
CX) b = 2a-5
Mixed Review
For Exercises 46 and 47, use the diagram at the right.
46. Measure DE and EFto the nearest millimeter.
47. Measure each angle to the nearest degree.
48. Draw a triangle that has sides of length 6 cm and 5 cm with a90° angle between those two sides.
Get Ready! To prepare for Lesson 1-2, do Exercises 49-51
Coordinate Geometry Graph the points on the coordinate plane.
49. (0, 0), (2, 2), (0, 3} 50. (1, 2), (-4, 3), (-5, 0)
^ See p. 884.
^ See p. 893.
51. (-4, -5), (0,-1), (3, -2)
10 Chapter 1 Tools of Geometry