Transcript Document

Rotations
Section 11.8
Goal
• Identify rotations and rotational symmetry.
Key Vocabulary
•
•
•
•
Rotation
Center of rotation
Angle of rotation
Rotational symmetry
Rotation Vocabulary
• Rotation – transformation that turns
every point of a pre-image through a
specified angle and direction about a
fixed point.
image
Pre-image
rotation
fixed point
Rotation Vocabulary
• Center of rotation – fixed point of the
rotation.
Center of Rotation
Rotation Vocabulary
• Angle of rotation – angle between a
pre-image point and corresponding
image point.
image
Angle of Rotation
Pre-image
Example:
Click the
triangle to
see rotation
Center of
Rotation
Example 1: Identifying Rotations
Tell whether each transformation appears to be a rotation.
Explain.
A.
B.
No; the figure appears
to be flipped.
Yes; the figure appears
to be turned around a point.
Your Turn:
Tell whether each transformation appears to be a rotation.
b.
a.
No, the figure appears to be a
translation.
Yes, the figure appears to be
turned around a point.
Rotation Vocabulary
• Rotational symmetry – A figure in a plane has
rotational symmetry if the figure can be mapped onto
itself by a rotation of 180⁰ or less.
Has rotational symmetry because it maps onto itself by a rotation of 90⁰.
Rotational Symmetry
When a figure can be rotated less than 360° and the
image and pre-image are indistinguishable (regular
polygons are a great example).
Symmetry
Rotational:
120°
90°
60°
45°
Example 2
Identify Rotational Symmetry
Does the figure have rotational symmetry? If so,
describe the rotations that map the figure onto itself.
a. Rectangle b. Regular hexagon c. Trapezoid
SOLUTION
a. Yes. A rectangle can be mapped onto itself by a
clockwise or counterclockwise rotation of 180°
about its center.
Example 2
Identify Rotational Symmetry
Regular hexagon
b. Yes. A regular hexagon can be mapped onto itself
by a clockwise or counterclockwise rotation of 60°,
120°, or 180° about its center.
Trapezoid
c. No. A trapezoid does not have rotational symmetry.
Your Turn:
Does the figure have rotational symmetry? If so,
describe the rotations that map the figure onto itself.
1. Isosceles trapezoid
ANSWER
no
2. Parallelogram
ANSWER
yes; a clockwise or counterclockwise
rotation of 180° about its center
Your Turn:
3. Regular octagon
ANSWER
yes; a clockwise or counterclockwise
rotation of 45°, 90°, 135°, or 180° about its
center
Rotating a Figure
Rotate quadrilateral VIKE through an angle of 130 abut point Q.
Draw segment VQ.
Draw circle Q with radius VQ.
K
Measure 130º clockwise with vertex at Q.
I
V
Mark point D and draw segment QD.
Extend segment QD to intersect circle.
E
Mark intersection V’.
You have just rotated V thru a ∠130º .
Q
D
V’
Rotating a Figure
Rotate quadrilateral VIKE through an angle of 130 abut point Q.
Draw segment EQ.
Draw circle Q with radius EQ.
K
Measure 130º clockwise with vertex at Q.
I
V
Mark point D and draw segment QD.
E
Extend segment QD to intersect circle.
Mark intersection E’.
You have just rotated E thru a ∠130º .
Q
V’
E’
Rotating a Figure
Rotate quadrilateral VIKE through an angle of 130 abut point Q.
Draw segment KQ.
Draw circle Q with radius KQ.
K
Measure 130º clockwise with vertex at Q.
I
V
Mark point D and draw segment QD.
Extend segment QD to intersect circle.
E
Mark intersection K’.
You have just rotated K thru a ∠130º .
Q
V’
E’
K’
Rotating a Figure
Rotate quadrilateral VIKE through an angle of 130 abut point Q.
Circle Q with radius VQ goes thru I.
Draw segment IQ.
K
Measure 130º clockwise with vertex at Q.
I
V
Mark point D and draw segment QD.
Extend segment QD to intersect circle.
E
Mark intersection I’.
You have just rotated I thru a ∠130º .
Q
V’
E’
I’
K’
Rotating a Figure
Rotate quadrilateral VIKE through an angle of 130 abut point Q.
Connect point V’ to point I’.
Connect point I’ to point K’.
K
Connect point K’ to point E’.
I
V
Connect point E’ to point V’.
E
Q
V’
E’
I’
K’
Example 3
Rotations
Rotate ∆FGH 50° counterclockwise
about point C.
SOLUTION
1. To find the image of point F, draw
CF and draw a 50° angle. Find F' so
that CF = CF'.
2. To find the image of point G, draw
CG and draw a 50° angle. Find G' so
that CG = CG'.
Example 3
Rotations
3. To find the image of point H, draw CH
and draw a 50° angle. Find H' so that
CH = CH'. Draw ∆F'G'H'.
Rotation in a Coordinate Plane
• For a Rotation, you need;
• An angle or degree of turn
– Eg 90° or a Quarter Turn
– E.g. 180 ° or a Half Turn
• A direction
– Clockwise
– Anticlockwise
• A Centre of Rotation
– A point around which Object rotates
y
A Rotation of 90° Anticlockwise about (0,0)
(x, y)→(-y, x)
8
7
6
C(3,5)
5
B’(-2,4)
4
C’(-5,3)
3
1
x
A(2,1)
x
A’(-1,2)
B(4,2)
2
–7 –6
–5 –4 –3
– 2 –1
-1
-2
-3
-4
-5
-6
1
2
3 4
5
6
7 8
x
y
A Rotation of 180° about (0,0)
(x, y)→(-x, -y)
8
7
6
C(3,5)
5
4
3
B(4,2)
2
1
–5 –4 –3
B’(-4,-2)
– 2 –1
x
–7 –6
x
-1
A’(-2,-1)
-2
-3
-4
C’(-3,-5)
-5
-6
A(2,1)
1
2
3 4
5
6
7 8
x
Rotation in a Coordinate Plane
Example 4
Rotations in a Coordinate Plane
Sketch the quadrilateral with vertices A(2, –2), B(4, 1),
C(5, 1), and D(5, –1). Rotate it 90° counterclockwise
about the origin and name the coordinates of the new
vertices.
SOLUTION
Plot the points, as shown in blue.
Use a protractor and a ruler to find the
rotated vertices.
The coordinates of the vertices of the image
are A'(2, 2), B'(–1, 4), C'(–1, 5), and D'(1, 5).
Checkpoint
Rotations in a Coordinate Plane
4. Sketch the triangle with vertices A(0, 0), B(3, 0), and
C(3, 4). Rotate ∆ABC 90° counterclockwise about the
origin. Name the coordinates of the new vertices A',
B', and C'.
ANSWER
A'(0, 0), B'(0, 3), C'(–4, 3)
Assignment
• Pg. 636 – 639; #1 – 41 odd