Lesson 9-1
Reflections
or
Flips
Objectives
• Draw reflected images
•
•
•
•
Across x-axis
Across y-axis
Across origin
Across line y = x
• Recognize and draw lines of symmetry and
points of symmetry
• How many lines of symmetry in a figure?
• How many points of symmetry in a figure?
Vocabulary
• Reflection – is a transformation representing a flip of
a figure; figure may be reflected in a point, a line, or
a plane.
• Isometry – a congruence transformation (distance,
angle measurement, etc preserved)
• Line of symmetry – line of reflection that the figure
can be folded so that the two halves match exactly
• Point of symmetry – midpoint of all segments
between the pre-image and the image; figure must
have more than one line of symmetry
Reflections
Across the x-axis
y
B’
A’
Multiply y coordinate by -1
Across the origin
A
B
x
B
C’
x
C’
A
B’
B
y
y
A’
A
C
C
Across the y-axis
x
C
KEY:
Equal
Distance
from
Reflection
Line
Multiply x coordinate by -1
Across the line y = x
y
B
A
B’
C
C’
A’
x
C’
B’
A’
Multiply both coordinates by -1
Interchange x and y coordinates
Common reflections in the coordinate
plane
Reflection
x-axis
y-axis
origin
y=x
Pre-image to (a, b)  (a, -b) (a, b)  (-a, b) (a, b)  (-a, -b) (a, b)  (b, a)
image
Find
Multiply y
Multiply x
Multiply both
Interchange
coordinates
coordinate
coordinate
coordinates by -1
x and y
by -1
by -1
coordinates
A line of symmetry is like a line of reflection.
The line of symmetry in a figure is a line where the
figure could be folded in half so that the two halves
match exactly
Draw the reflected image of quadrilateral WXYZ in
line p.
Step 1
Draw segments
perpendicular to line p from each
point W, X, Y, and Z.
Step 2
Locate W', X', Y', and Z'
so that line p is the perpendicular
bisector of
Points W', X', Y', and Z' are the
respective images of W, X, Y, and
Z.
Step 3
Connect vertices W',
X', Y', and Z'.
Answer: Since points W', X', Y', and
Z' are the images of points W, X, Y,
and Z under reflection in line p, then
quadrilateral W'X'Y'Z' is the
reflection of quadrilateral WXYZ in
line p.
Draw the reflected image of quadrilateral ABCD in line n.
Answer:
COORDINATE GEOMETRY Quadrilateral ABCD has vertices A(1, 1),
B(3, 2), C(4, –1), and D(2, –3). Graph ABCD and its image under
reflection in the x-axis. Compare the coordinates of each vertex with
the coordinates of its image.
Use the vertical grid lines to find the corresponding point for each
vertex so that the x-axis is equidistant from each vertex and its
image.
D'
A(1, 1)  A' (1, –1)
C'
B(3, 2)  B' (3, –2)
C(4, –1)  C' (4, 1)
D(2, –3)  D' (2, 3)
A'
B'
Answer: The x-coordinates stay the same, but the
y-coordinates are opposite. That is, (a, b)  (a, –b).
COORDINATE GEOMETRY Quadrilateral ABCD has vertices A(1, 1),
B(3, 2), C(4, –1), and D(2, –3). Graph ABCD and its image under
reflection in the y-axis. Compare the coordinates of each vertex with
the coordinates of its image.
Use the horizontal grid lines to find the corresponding point for
each vertex so that the y-axis is equidistant from each vertex and
its image.
B'
A(1, 1)  A' (–1, 1)
A'
B(3, 2)  B' (–3, 2)
C(4, –1)  C' (–4, –1)
D(2, –3)  D' (–2, –3)
C'
D'
Answer: The x-coordinates are opposite, but the y-coordinates
stay the same. That is, (a, b)  (–a, b).
COORDINATE GEOMETRY Suppose quadrilateral ABCD with A(1, 2),
B(3, 5), C(4, –3), and D(2, –5) is reflected in the origin. Graph ABCD
and its image under reflection in the origin. Compare the
coordinates of each vertex with the coordinates of its image.
Use the horizontal and vertical distances from each vertex to the origin
to find the coordinates of its image. From A to the origin is 2 units down
and 1 unit left. A' is located by repeating that pattern from the origin.
A(1, 2)  A' (–1, –2)
B(3, 5)  B' (–3, –5)
D'
C'
C(4, –3)  C' (–4, 3)
D(2, –5)  D' (–2, 5)
Plot the reflected vertices and connect to
form the image A'B'C'D'. Comparing
coordinates shows that (a, b)  (–a, –b).
Answer: Both the x- and y-coordinates are
opposite. That is, (a, b)  (–a, –b).
A'
B'
COORDINATE GEOMETRY Suppose quadrilateral ABCD with A(1, 2),
B(3, 5), C(4, –3), and D(2, –5) is reflected in the line y = x. Graph
ABCD and its image under reflection in the line y = x. Compare the
coordinates of each vertex with the coordinates of its image.
The slope of y = x is 1. AA’ is perpendicular to y = x so its slope is –1. From A
to the line y = x move down ½ unit and right ½ unit. From the line y = x move
down ½ unit, right ½ unit to A'.
C'
A(1, 2)  A'(2, 1)
B(3, 5)  B'(5, 3)
C(4, –3)  C'(–3, 4)
B'
D'
A'
D(2, –5)  D'(–5, 2)
Plot the reflected vertices and
connect to form the image
A'B'C'D'.
Answer: The x-coordinate becomes the y-coordinate and the ycoordinate becomes the x-coordinate. That is, (a, b)  (b, a).
BILLARDS
Dave challenged Juan to hit
the 8 ball in the left corner
pocket. Describe how Juan
should hit the ball using
reflections.
Answer:
Juan should mentally reflect
the left corner pocket in the
line that contains the right
side of the table. If he hits
the ball at the reflected
image of the pocket, the ball
will strike the right side and
rebound on a path toward
the left corner pocket.
Determine how many lines of symmetry a regular
pentagon has. Then determine whether a regular
pentagon has a point of symmetry.
A regular pentagon has five lines of symmetry.
A point of symmetry is a point that is a common point of
reflection for all points on the figure. There is not one point
of symmetry in a regular pentagon.
Answer: 5; no
a. Determine how many lines of symmetry an equilateral
triangle has. Then determine whether an equilateral
triangle has a point of symmetry.
Answer: 3; no
b. Determine how many lines of symmetry a hexagon
has. Then determine whether a hexagon has a point of
symmetry.
Answer: 6; yes
Summary & Homework
• Summary:
Reflection
x-axis
y-axis
origin
y=x
Pre-image to (a, b)  (a, -b) (a, b)  (-a, b) (a, b)  (-a, -b) (a, b)  (b, a)
image
Find
Multiply y
Multiply x
Multiply both
Interchange
coordinates
coordinate
coordinate
coordinates by -1
x and y
by -1
by -1
coordinates
– Line of Symmetry – a line across which the figure
could be folded in half
– Point of Symmetry – even numbered regular
figures only for us
• Homework:
– Take notes on this PPT (Power Point )