ch 3 spiral review. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16
TRANSCRIPT
Ch 3 Spiral Review
1.
2.
3.
4.
5.
6.
7.
8.
∠DEG =28°
4 2
2x2 −2x−4
B
−4.5,5( )
37°
x =16
no,curved
9.
10.
11.
12.
13.
14.
15.
16.
yes,concave
C
∠AFB& ∠EFD
∠BFC& ∠CFD
65
11 + 65
A
D
17.
18.
19.
True
false
B
Homework Scoring
• Count how many problems you missed or didn’t do
• 0-1 missed = 10• 2-3 missed = 9• 4-5 missed = 8• 6-7 missed = 7• 8-9 missed = 6
• 10-11 missed = 5• 12-13 missed = 4• 14-15 missed = 3• 16-17 missed = 2• 18-19 missed = 1• 20-21 missed = 0
– Similarity and Congruence
Measuring, describing, and transforming: these are three major skills in geometry that you have been developing. In this chapter, you will focus on comparing; you will explore ways to determine if two figures have the same shape (called similar) and if they have the same size (congruent). Making logical and convincing arguments that support specific ideas about the shape you are studying is another important skill. In this chapter you will learn how you can document facts to support a conclusion in a flowchart and two-column proof.
4.1
What Do These Shapes Have In common?
Pg. 3Similarity
4.1 – What do These Shapes Have in Common?Similarity
This year you have performed transformations on shapes that were congruent. But what makes two figures look alike? Today you will be introduced to a new transformation that enlarges or shrinks a figure while maintaining its shape, called a dilation. After creating new enlarged shapes, you and your team will explore the interesting relationships that exist between figures that have the same shape.
4.1 – WARM-UP STRETCHBefore computers and copy machines existed, it sometimes took hours to enlarge documents or to shrink text on items such as jewelry. A pantograph device (like the one at right) was often used to duplicate written documents and artistic drawings. You will now employ the same geometric principles by using rubber bands to draw enlarged copies of a design. Your teacher will show you how to do this.
Corresponding angles are equal
Corresponding sides are bigger
a. What do you notice about the angles of the original and the dilation? b. What do you notice about the sides of the original and the dilation?
4.2 – DILATIONIn problem 4.1, you created designs that were similar, meaning that they have the same shape. But how can you determine if two figures are similar? What do similar shapes have in common? To find out, your team will need to create similar shapes that you can measure and compare.
• Resource Manager: Only one that can ask teacher a question (does 2x)
• Facilitator: Checks to make sure everyone understands, takes turns reading (does 3x)
• Recorder/Reporter: Makes sure everyone knows how to draw (does 4x)
• Team Captain: Keeps group on task, fills in for missing students, keeps track of time (does 5x)
(3,2)(6,4)
(9,6)
(12,8)
B’
C’ D’
b. Carefully cut out your enlarged shape and compare it to your teammates' shapes. How are the four shapes different? How are they the same? As you investigate, make sure you record what qualities make the shapes different and what qualities make the shapes the same. Then complete the conditional statement.
If a shape is similar, then: • ____________________________
• ____________________________
Corresponding angles are equal
Corresponding sides are proportional
4.3 – ZOOM FACTOR AND SCALE FACTORIn the previous problem, you learned that you can create similar shapes by multiplying each side length by the same number. This number is called the zoom factor or the scale factor. You may have used a zoom factor when using a copy machine. For example, if you set the zoom factor on a copier to 50%, the machine shrinks the image in half (that is, multiplies by 0.5) but keeps the shape the same. In this course, the zoom factor and the scale factor will be used to describe the ratio of the new figure to the original.
The dashed-line figure is a dilation image of the solid-line figure. The labeled point is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation.
12
2015
= 43
84
= 21
4.4 – SCALE FACTORUse the given scale factor for the given shapes. When does the shape become bigger? When does it shrink?
4.5 – MULTIPLY VS. ADDCan you create a similar shape if you add the same number to each side? Examine this again with the rectangle at right. When does this shape appear to be similar to the original?
5
3
5
3 15
9
x3
Similar when multiplying
5
3
8
6
+3
Not similar when adding
4.6 – PICTURES Draw an example of a shape that is similar and congruent to the following.
Similar:
Congruent:
4.7 – SYMBOLS Talk about the differences of each of the symbols below. What do you think each one is written the way it is? How are they alike? How are they different?
=Exactly the same Close Estimate
~Alike, but not equal Objects w/ same
size and shape
a.Equal to b. Approximately
c. Similar d. Congruent
4.8 – CONGRUENT VS. EQUAL When you are talking about numbers that are the same, we can say they are equal. However, when it is a shape, we call it congruent, because it isn't exactly the same. Use this idea to complete the statements below.
BC 8in
C∠ 45
4.9 – DEFINITIONS
Ratio
Scale Factor
Relation between 2 numbers
a:b ab
a to b
Always Reduce!
The enlargement or reduction in shape
4.10 – RATIOSUse the number line to find the ratio of the distances.
AC
AB 26
= 13=
4.10 – RATIOSUse the number line to find the ratio of the distances.
23
=CD
AB
4.10 – RATIOSUse the number line to find the ratio of the distances.
144
= 72
=BC
AE
4.10 – RATIOSUse the number line to find the ratio of the distances.
142
= 71
=AB
AE
4.11 – RATIOS TO SOLVEa. The measures of two supplementary angles are in the ratio 4:5. What is the measure of the smaller angle?
4x + 5x = 180
9x = 180x = 20
Smallest: 4(20) = 80°
b. The measures of the angles in a triangle are in the ratio 5:6:7. What is the measure of the smallest angle?
6x + 7x = 180
18x = 180x = 10
Smallest: 5(10) = 50°
5x +
Scale Me Up
• Each person in your group must bring in a cartoon tomorrow!