no.8 2.5 Day 1 Notes

#8 GEOMETRY Name________________________ SECTION 2.5 DAY 1 NOTES – ROTATIONS Date_______Class Period_______ __________________________________________________________________________________________
Target Goals: ‐ Identify a rotation of a point about a fixed point in a plane ___Great ___OK ___  ‐ Determine coordinates of a rotated point in a plane ___Great ____OK ___  __________________________________________________________________________________________ A'
A
B'
A rotation is an isometric transformation that turns a figure about a fixed point called the center of rotation. Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation
(notation Rcenter, degree). An object and its rotation are the same shape and size, but the figures may be turned in different directions. B
C
θ
C'
O
RO ,  ABC   A ' B ' C ' A'
A'
A
B'
A
B'
A'
A
B'
C
C'
θ
O
B
C
θ
C'
O
B
C
C'
B
θ
O
What is true about OA and OA ', OB and OB ', OC and OC '? _______________ How do you know this is true? _______________________________________________ ROTATIONAL DIRECTION: One full rotation is 360o, this would return all points in the plane to their original location. Because a rotation can go in two directions along the same arc we need to define positive and negative rotation values. COUNTERCLOCKWISE IS A POSITIVE DIRECTION, and clockwise is a negative direction. SPECIAL ROTATION – ROTATION OF 180 A rotation of 180o maps A to A’ such that:
a) mAOA’ = 180 o (from definition of rotation) b) OA = OA’ (from definition of rotation) A
O


c) Ray OA and Ray OA ' are opposite rays. (They form a line.) 

AO is the same line as AA ' A'
Samples: 1. For #1 and #2, use the grid or patty paper to rotate the following figures. Label the image! A
B RO , 900 ( ABC ) A = (_____, _____)  A’ = (_____, _____) C RO , 900 ( ABC ) B = (_____, _____)  B’ = (_____, _____) RO , 900 ( ABC ) C = (_____, _____)  C’ = (_____, _____) 2. A B RO , 1800 ( ABCD ) A = (_____, _____)  A’ = (_____, _____) RO , 1800 ( ABCD ) B = (_____, _____)  B’ = (_____, _____) D C RO , 1800 ( ABCD ) C = (_____, _____)  C’ = (_____, _____) RO , 1800 ( ABCD ) D = (_____, _____)  D’ = (_____, _____) 3. Determine the point that meets the given criteria. a) RP ,  450 ( A)  ______ G
c) RP , 1800 (_____) 
F d) RP , 450 (_____) 
P e) RP , 2250 (G )  ______ f) RP , 1350 (______)  G C F
E Rules for Rotations: B
45° A
E H b) RP , 900 (C )  ______ D
ROTATION ON THE COORDINATE GRID 4
Rotation of 90 about the Origin C' (-y,x)
B (2,4)
4
C' (-2,3)
When we rotate 90 about the origin, we see that the x and y B' (-4,2)
coordinates are reversed and the new x coordinate is negated. 5
C (x,y)
2
2
C (3,2)
A' (-1,1)
A (1,1)
RULE FOR ROTATION BY 90 ABOUT THE ORIGIN RO ,90 ( x, y )  ( y, x) Rotation of 180 about the Origin When we rotate by 180 about the origin, we see the x and y coordinates are negated. 4
B (2,4)
C (3,2)
2
2
C (x,y)
A (1,1)
A' -(1,-1)
C' (-3,-2)
2
2
C' (-x,-y)
'B (-2-,4)
4
RULE FOR ROTATION BY 180 ABOUT THE ORIGIN RO ,180 ( x, y )  ( x,  y ) Rotation of 270 about the Origin 4
B (2,4)
This is also a rotation of ‐ 90. 2
2
C (x,y)
C (3,2)
When we rotate by 270 about the origin, we see that the x and y coordinates are reversed and the new y coordinate is negated. A (1,1)
5
A' (1,-1)
2
B' (4,-2)
C' (y,-x)
2
C' (2,-3)
RULE FOR ROTATION BY 270 ABOUT THE ORIGIN RO ,270 ( x, y )  ( y,  x) Assignment #8 Worksheet

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