Transcript Slide 1

5.7 Reflections and Symmetry
Objective
•
Identify and use reflections and lines of
symmetry.
Key Vocabulary
•
•
•
•
Preimage
Image
Reflection
Line of Symmetry
Identifying Transformations
• Figures in a plane can be
– Reflected
– Translated – (did section 3.7)
– Rotated – (later)
• To produce new figures. The new figure is
called the IMAGE. The original figure is
called the PREIMAGE. The operation that
MAPS, or moves the preimage onto the
image is called a transformation.
Transformation
Definition – anything that maps (or moves) an
original geometric figure, the preimage, onto a
new figure called the image.
Some facts
• Some transformations involve labels. When
you name an image, take the corresponding
point of the preimage and add a prime
symbol. For instance, if the preimage is A,
then the image is A’, read as “A prime.”
Mapping
Preimage
Original shape or
object.
Image
Shape or object
after it has been
moved.
A
A’
Reflection (flip)
• A reflection or flip is a
transformation over a line
called the line of reflection.
Each point of the preimage
and its image are the same
distance from the line of
reflection.
A
A’
Reflection (flip)
The result of a figure
flipped over a line of
reflection.
Click on this
trapezoid to
see
reflection.
The result of a figure flipped over a line of reflection.
Reflections
• Look at yourself in a mirror!
– How does your reflection respond as you step
toward the mirror? away from the mirror?
• When a figure is reflected, the image is
congruent to the original.
• The actual figure and its image appear the
same distance from the line of reflection,
here the mirror.
Example 1: Identifying Reflections
Tell whether each transformation appears to be a reflection.
Explain.
B.
A.
No; the image does not
Appear to be flipped.
Yes; the image appears to be
flipped across a line..
Your Turn
Tell whether each transformation appears to be a
reflection.
a.
b.
No; the figure does not
appear to be flipped.
Yes; the image appears to
be flipped across a line.
Properties
Example 2
Identify Reflections
Tell whether the red triangle is
the reflection of the blue
triangle in line m.
SOLUTION
Check to see if all three properties of a reflection are met.
1. Is the image congruent to the original figure? Yes.
2. Is the orientation of the image reversed? Yes.
∆FGH has a clockwise orientation.
∆F'G'H' has a counter clockwise orientation.
3. Is m the perpendicular bisector of the segments
connecting the corresponding points? Yes.
Example 2
Identify Reflections
To check, draw a diagram
and connect the
corresponding endpoints.
ANSWER
Because all three properties are met, the
red triangle is the reflection of the blue
triangle in line m.
Example 3
Identify Reflections
Tell whether the red triangle is
the reflection of the blue triangle
in line m.
SOLUTION
Check to see if all three properties of a reflection are met.
1. Is the image congruent to the original figure? Yes.
2. Is the orientation of the image reversed? No.
ANSWER
The red triangle is not a reflection of the
blue triangle.
Drawing Reflections
Copy the triangle and the line of reflection. Draw the
reflection of the triangle across the line.
Step 1 Through each vertex draw a line perpendicular to the
line of reflection.
Drawing Reflections Continued
Step 2 Measure the distance from each vertex to the line of
reflection. Locate the image of each vertex on the opposite
side of the line of reflection and the same distance from it.
Drawing Reflections Continued
Step 3 Connect the images of the vertices.
To graph the reflection of point
A(1,-2) over the y-axis:
Reflections in a
Coordinate Plane
1.Identify the y-axis as the line of reflection
(the mirror).
2. Point A is 1 unit to the right of the y-axis,
so its reflection A’ is 1 unit to the left of
the y-axis.
We discovered an interesting phenomenon:
simply change the sign of the xcoordinate to reflect a point over the yaxis!
Similarly, to graph the reflection of a point over the x-axis, simply change
the sign of the y-coordinate!
Example 4
Reflections in a Coordinate Plane
a. Which segment is the reflection
of AB in the x-axis? Which point
corresponds to A? to B?
b. Which segment is the
reflection of AB in the y-axis?
Which point corresponds to
A? to B?
SOLUTION
a. The x-axis is the perpendicular bisector of AJ and BK,
so the reflection of AB in the x-axis is JK.
A(–4, 1)
J(–4, –1)
A is reflected onto J.
B(–1, 3)
K(–1, –3)
B is reflected onto K.
Example 4
Reflections in a Coordinate Plane
Which segment is the
reflection of AB in the y-axis?
Which point corresponds to
A? to B?
b. The y-axis is the perpendicular bisector of AD and
BE, so the reflection of AB in the y-axis is DE.
A(–4, 1)
D(4, 1)
A is reflected onto D.
B(–1, 3)
E(1, 3)
B is reflected onto E.
Your Turn:
Identify Reflections
Tell whether the red figure is a reflection of the blue
figure. If the red figure is a reflection, name the line
of reflection.
3.
2.
1.
ANSWER
yes; the
x-axis
ANSWER
ANSWER
no
yes; the
y-axis
Reflecting a Figure
• Use these same steps to reflect an entire
figure on the coordinate plane:
– Identify which axis is the line of reflection (the
mirror).
– Individually reflect each endpoint of the
figure.
– Connect the reflected points.
What letter would
you get if you
reflected each
shape in its
corresponding
mirror line?
What letter would
you get if you
reflected each
shape in its
corresponding
mirror line?
What letter
would you get if
you reflected
each shape in its
corresponding
mirror line?
What letter
would you get if
you reflected
each shape in its
corresponding
mirror line?
What letter would
you get if you
reflected each
shape in its
corresponding
mirror line?
What letter would
you get if you
reflected each
shape in its
corresponding
mirror line?
Now it’s your turn…
• On your worksheet, reflect every shape in
the corresponding mirror line.
• Use tracing paper to help you.
• All the shapes should fit together to form a
word.
• Draw in pencil in case you make any
mistakes.
Symmetry
What is a line of symmetry?
• A line on which a figure
can be folded so that both
sides match.
• A line of symmetry is a line
of reflection.
• A figure may have more
than one line of symmetry.
Real-World Symmetry
Connection
• Line symmetry can be found in works of art and in nature.
Some figures, like this tree,
have only one line of symmetry.
Vertical symmetry
Others, like this snowflake,
have multiple lines of symmetry.
Vertical, horizontal, & diagonal symmetry
Which of these flags have
a line of symmetry?
United States of America
Maryland
Canada
England
What about these
math symbols?
Do they have symmetry?
Example 5
Determine Lines of Symmetry
Determine the number of lines of symmetry in a square.
SOLUTION
Think about how many different ways you can fold a
square so that the edges of the figure match up perfectly.
vertical fold horizontal fold diagonal fold diagonal fold
ANSWER
A square has four lines of
symmetry.
Example 6
Determine Lines of Symmetry
Determine the number of lines of symmetry in each
figure.
a.
b.
c.
b. no lines of
symmetry
c. 6 lines of
symmetry
SOLUTION
a. 2 lines of
symmetry
How many lines of symmetry to
these regular polygons have?
Do you
see a
pattern?
Example 7
Use Lines of Symmetry
Mirrors are used to create images seen through a
kaleidoscope. The angle between the mirrors is A.
Find the angle measure used to create the
180°
kaleidoscope design. Use the equation mA = n ,
where n is the number of lines of symmetry
in the pattern.
a.
b.
c.
Example 7
Use Lines of Symmetry
SOLUTION
a. The design has 3 lines of symmetry. So, in the
formula, n = 3.
180° 180°
mA = n =
= 60°
3
b. The design has 4 lines of symmetry. So, in the
formula, n = 4.
180° 180°
mA = n =
= 45°
4
c. The design has 6 lines of symmetry. So, in the
formula, n = 6.
180° 180°
mA = n =
= 30°
6
Your turn:
Determine Lines of Symmetry
Determine the number of lines of symmetry in the
figure.
1.
2.
ANSWER
1
ANSWER
2
ANSWER
4
3.
Assignment
• Pg. 286-290 : #1 – 39 odd, 43 – 49 odd.