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Aim 13d rotations in the coordinate plane.notebook
1
December 07, 2015
Dec 311:11 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
Do Now: Use the examples below to complete the rules for Rotations whose center is the origin.
-y x
y -x
-x -y
Remember that a rotation of 90 ccw gets you to the same place as 270 cw
Remember that a rotation of 180 ccw gets you to the same place as 180 cw
Remember that a rotation of 270 ccw gets you to the same place as 90 cw
5 19 1**
5 -5-1 -5-1 7
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
2 3
2 5
4 6
5 4
4 2
-3 2
-5 2
-6 4
-4 5
-2 4
P'P''
Q'
v
Q''
R'
R'' S'
S''
T'
T''
ccw
Aim 13d rotations in the coordinate plane.notebook
2
December 07, 2015
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
4. CHALLENGE: Point C has coordinates C(5, 5). The image of this point after a rotation of 100° about a certain point is C′(5, 7.5). Find the coordinates of the center of this rotation. (Hint: To be precise use each box as )
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
6
7
8
c
c'
0°249
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
4. CHALLENGE: Point C has coordinates C(5, 5). The image of this point after a rotation of 100° about a certain point is C′(5, 7.5). Find the coordinates of the center of this rotation. (Hint: To be precise use each box as )
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
6
7
8
c
c'
0°249
Aim 13d rotations in the coordinate plane.notebook
3
December 07, 2015
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
4. CHALLENGE: Point C has coordinates C(5, 5). The image of this point after a rotation of 100° about a certain point is C′(5, 7.5). Find the coordinates of the center of this rotation. (Hint: To be precise use each box as )
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
6
7
8
c
c'
There are 3 points you can guess. Now make the angle and check which one is 100 degrees
0°249
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
4. CHALLENGE: Point C has coordinates C(5, 5). The image of this point after a rotation of 100° about a certain point is C′(5, 7.5). Find the coordinates of the center of this rotation. (Hint: To be precise use each box as )
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
6
7
8
c
c'
The coordinate (-1,2) makes and angle of rotation of 100 degrees.
0°249
0180
10
17020
16030
15040
140
50
130
60
120
70
110
80
100
90
90
27°
100
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110
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120
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13050
14040
15030
16020
17010
180
0
Aim 13d rotations in the coordinate plane.notebook
4
December 07, 2015
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
4. CHALLENGE: Point C has coordinates C(5, 5). The image of this point after a rotation of 100° about a certain point is C′(5, 7.5). Find the coordinates of the center of this rotation. (Hint: To be precise use each box as )
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
6
7
8
c
c'
There are 3 points you can guess. Now make the angle and check which one is 100 degrees
0°249
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
5. Are collinearity and betweenness of points maintained under rotation? (You can make up an example with the extra space on the graph.)
Explain.Collinear points are points on the same line. So if I rotate collinear points will the image points still be colliner? Betweenness looks like the partition postulate. For example, If AB+BC=AC , then B is between A and C. So rotate 90 degrees and If A'B'+B'C'=A'C'then will B' still be between A' and C'? The answers to both questions is YES, since rotations are a rigid motion which produces a congruent shape. So the segments drawn in each case will produce congruent segments when rotated.
Aim 13d rotations in the coordinate plane.notebook
5
December 07, 2015
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
For queson 7 & 8, Locate the center of rotation P, then find the angle of rotation.6)
point is center of rotation since it is invariant (does not move).
P
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
For queson 7 & 8, Locate the center of rotation P, then find the angle of rotation.6)
Rotation ccw 82°P
0
180
10
170
20
160
30
150
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140
50
130
60
120
70
110
80100
909082°
10080
11070
12060
130
50 140
40 150
30160
20
170
10
180
0
P,82°R
Aim 13d rotations in the coordinate plane.notebook
6
December 07, 2015
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
For queson 7 & 8, Locate the center of rotation P, then find the angle of rotation.7)
0°
208
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
For queson 7 & 8, Locate the center of rotation P, then find the angle of rotation.7)
0° 168
Aim 13d rotations in the coordinate plane.notebook
7
December 07, 2015
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
For queson 7 & 8, Locate the center of rotation P, then find the angle of rotation.7)
•P0° 168
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
For queson 7 & 8, Locate the center of rotation P, then find the angle of rotation.7)
•P0° 168
Aim 13d rotations in the coordinate plane.notebook
8
December 07, 2015
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
For queson 7 & 8, Locate the center of rotation P, then find the angle of rotation.7) cw,
obtuse122°
0° 168
0180
10 170
20 160
30150
40140
50
13060
12070
11080
100
90
90
18°
100
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110
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50
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150
30
160
20
170
10
1800
RP,122°•P
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
Use a protractor and ruler to draw a rotation of the figure the given number of degrees about K.
0°86
0
180
10
170
20
160
30
150
40
140
50
130
60
120
70
110
80
100
90
90
119° 10080
110 70
120 60
13050
14040
150
30160
20170
10180
0
Aim 13d rotations in the coordinate plane.notebook
9
December 07, 2015
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
Use a protractor and ruler to draw a rotation of the figure the given number of degrees about K.
0°86
0
180
10
170
20
160
30
150
40
140
50
130
60
120
70
110
80
100
90
90
119° 10080
110 70
120 60
13050
14040
150
30160
20170
10180
0
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
Use a protractor and ruler to draw a rotation of the figure the given number of degrees about K.
•
0°86
0
180
10
170
20
160
30
150
40
140
50
130
60
120
70
110
80
100
90
90
119° 10080
110 70
120 60
13050
14040
150
30160
20170
10180
0
Aim 13d rotations in the coordinate plane.notebook
10
December 07, 2015
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
Use a protractor and ruler to draw a rotation of the figure the given number of degrees about K.
•
0°86
0
180
10
170
20
160
30
150
40
140
50
130
60
120
70
110
80
100
90
90
119° 10080
110 70
120 60
13050
14040
150
30160
20170
10180
0
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
Use a protractor and ruler to draw a rotation of the figure the given number of degrees about K.
••
T'
0°86
0
180
10
170
20
160
30
150
40
140
50
130
60
120
70
110
80
100
90
90
119° 10080
110 70
120 60
13050
14040
150
30160
20170
10180
0
Aim 13d rotations in the coordinate plane.notebook
11
December 07, 2015
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
Use a protractor and ruler to draw a rotation of the figure the given number of degrees about K.
• •T'
0° 100
0
180
10
170
20
160
30
150
40140
50 130
60 120
70110
80100
90
90
57°
100
80110
70120
60
130
50
140
40
150
30
160
20
170
10
180
0
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
Use a protractor and ruler to draw a rotation of the figure the given number of degrees about K.
••
T'
0° 100
0
180
10
170
20
160
30
150
40
140
50
130
60
120
70
110
80
100
90
90
119° 10080
110 70
120 60
13050
14040
150
30160
20170
10180
0
Aim 13d rotations in the coordinate plane.notebook
12
December 07, 2015
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
Use a protractor and ruler to draw a rotation of the figure the given number of degrees about K.
•
• •T'
0° 100
0180
10
17020
16030
15040
140
50
130
60
120
70
110
80
100
90
90
21°
100
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110
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120
60
130
50
14040
15030
16020
17010
1800
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
Use a protractor and ruler to draw a rotation of the figure the given number of degrees about K.
• •
•
T'
0° 100
0180
10
17020
16030
15040
140
50
130
60
120
70
110
80
100
90
90
21°
100
80
110
70
120
60
130
50
14040
15030
16020
17010
1800
Aim 13d rotations in the coordinate plane.notebook
13
December 07, 2015
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
Use a protractor and ruler to draw a rotation of the figure the given number of degrees about K.
•
•
•
•
R'
T'
0° 100
0180
10
17020
16030
15040
140
50
130
60
120
70
110
80
100
90
90
21°
100
80
110
70
120
60
130
50
14040
15030
16020
17010
1800
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
Use a protractor and ruler to draw a rotation of the figure the given number of degrees about K.
•
•
•
R'
T'
0°148
0
180
10
170
20
160
30
150
40140
50 130
60 120
70110
80100
90
90
56°
100
80110
70120
60
130
50
140
40
150
30
160
20
170
10
180
0
Aim 13d rotations in the coordinate plane.notebook
14
December 07, 2015
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
Use a protractor and ruler to draw a rotation of the figure the given number of degrees about K.
•
•
•
R'
T'0°
148
0
180
10
170
20
160
30
150
40140
50 130
60 120
70110
80100
90
90
56°
100
80110
70120
60
130
50
140
40
150
30
160
20
170
10
180
0
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
Use a protractor and ruler to draw a rotation of the figure the given number of degrees about K.
•
•
•
•
R'
T'0°
148
0
180
10170
20160
30150
40140
50130
60
12070
110 80
10090
90
151°
100
80
110
70
120
60
130
50
140
40
150
30
160
20
170
10
180
0
Aim 13d rotations in the coordinate plane.notebook
15
December 07, 2015
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
Use a protractor and ruler to draw a rotation of the figure the given number of degrees about K.
•
•
•
•
R'
T'0°
148
0
180
10170
20160
30150
40140
50130
60
120 70
110 80
10090
90
151°
100
80
110
70
120
60
130
50
140
40
150
30
160
20
170
10
180
0
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
Use a protractor and ruler to draw a rotation of the figure the given number of degrees about K.
•
•
•
•
•S'
0°148
0
180
10170
20160
30150
40140
50130
60
12070
110 80
10090
90
151°
100
80
110
70
120
60
130
50
140
40
150
30
160
20
170
10
180
0
R'
T'
Aim 13d rotations in the coordinate plane.notebook
16
December 07, 2015
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
Use a protractor and ruler to draw a rotation of the figure the given number of degrees about K.
•
•
•
•
•S'
0°148
0
180
10170
20160
30150
40140
50130
60
120 70
110 80
10090
90
151°
100
80
110
70
120
60
130
50
140
40
150
30
160
20
170
10
180
0
R'
T'
Dec 311:12 PM
Aim 13d: How do we rotate coordinates on a graph (rules)?
10. Def: An invariant point of a function is an element of the function's domain that is mapped to itself. Do invariant points sometimes, always, or never occur in a rotation? Explain your reasoning.
If the center of rotation is a point on the shape being rotated, then it does not move. Therefore it is an invariant point.