Aim #15: What types of symmetry does a figure have?
CC Geometry H
Do Now: ΔA has been reflected across line x, and the image is ΔB. With a
compass, construct ΔC, the image of ΔB after it is reflected across line y.
A student looked at ΔC and said, "That's not the
image of ΔA after two reflections - that's
the image of ΔA after a rotation!" Do you agree?
Where would the center of rotation be?
y
B
A
x
Line x and line y are each a _____ __ _____________. When a figure is
reflected across a line,and the resulting image is reflected across a second line
which intersects thefirst line, the final image is a ______________ of the original
figure. Name the angle ofrotation for the above example in 3 ways:
A figure has symmetry if there exists a rigid motion (e.g. reflection, rotation)
that maps the figure back onto itself.
LINE SYMMETRY OF A FIGURE
A figure has line symmetry if the figure can be mapped onto itself by a reflection
in a line, called the line of symmetry. Every point on one side of the line has a
corresponding point on the other side of the line, and the line is equidistant from
the corresponding pairs of points.
1) Draw in all lines of symmetry for each object.
C
parallelogram
rectangle
square
2) Construct a line of symmetry for isosceles ΔABC. (AB = CB)
B
81 153°
A
C
Regular Polygon: A polygon is regular if all sides have equal length (is
equilateral) and all interior angles have equal measure (is equiangular).
3) For the regular hexagon below, construct one line of symmetry that
does not intersect a vertex. Then sketch in all remaining lines of symmetry.
14
0°
ROTATIONAL AND IDENTITY SYMMETRY OF A FIGURE
A figure has rotational symmetry if the figure can be mapped onto itself by a
rotation between 0˚ and 360˚ about the center of the figure, called the
center of rotation. A figure has identity symmetry, I, if each point in the figure is
mapped back onto itself. All figures have identity symmetry. When listing
o
symmetries of a figure, you may include 360 rotational symmetry which is the
identity symmetry.
A regular polygon can be formed by duplicating and rotating an isosceles triangle.
The isosceles triangle below is rotated ___ times to get the triangle back to its
o
original position. Since 360 is a full rotation, ____ ÷ ____ gives theminimum
number of degrees for the regular octagon to map a regular octagon onto itself.
4) What is the minimum number of degrees of rotation that maps a regular
decagon onto itself?
o
5) If you repeatedly rotated a triangle 12 each time to create a regular polygon,
how many sides would this polygon have?
6) If you repeatedly rotated a triangle to create a regular hexagon, how many
degrees would you rotate the triangle each time?
all
7) Determine the figures below that have rotational symmetry and, if so, name
angles of rotation.
parallelogram
square
rectangle
regular
pentagon
8) Construct the center of rotation for each regular polygon.
104
125°
9) In the equilateral triangle to the right with
o
circumcenter D, ______ rotations of _____
occurred.
D
10) ABCD is a square.
a) Draw all lines of symmetry. How many are there? Label the center of rotational
symmetry.
A
B
a) b) Describe what kinds of symmetry the square has.
o
D
c) How many are rotations? (include 360 , the identity symmetry.)
d) How many are reflections?
C
11) Shade two of the nine smaller squares so the resulting figure has:
only one vertical
and one horizontal
line of symmetry
only one horizontal
line of symmetry
only two lines of
symmetry about the
diagonals
12) James says that the figure has only line symmetry.
Jewel says that the figure has only rotational
symmetry. Is either of them correct? Explain.
Let's sum it up!!
• A line of symmetry divides a figure into two congruent halves that are
reflections of each other.
• If a figure has rotational symmetry, it can be rotated about the center of
o
o
rotation between 0 and 360 to perfectly overlap itself.
• The minimum angle of rotation that makes a regular polygon identical to the
original polygon is 360/n (n = number of sides).
• If a figure has identity symmetry, each point is mapped onto itself.
Name _____________________
Date _____________________
CC Geometry H
HW #15
1. Circle figures with line symmetry. Underline figures with rotational symmetry,
o
and list all angles of rotation, including 360 , the identity symmetry.
NOON
A
2. ABCDE is a regular pentagon.
a) Draw all lines of symmetry . Label S, the center of
rotational symmetry.
E
B
b) List all kinds of symmetry.
C
D
c) How many are rotations (including identity symmetry)?
d) How many are reflections?
e) To how many places can vertex A be moved by a symmetry of the pentagon?
3. Draw a figure, not previously in Aim 15, that has line but not rotational symmetry.
4. Shade two of the nine smaller squares so the resulting figure has:
only one line of symmetry
about a diagonal
no line of symmetry
OVER
4. Consider all the symmetries of the following figure.
a. How many are rotations (including the identity symmetry)?
b. How many are reflections?
c. How could you shade the figure so that the resulting figure only has 3
possible rotational symmetries (including the identity symmetry)?
Review
5. Find the measure of each angle of the triangle.
2y+
40
6y 2x+20
y+20
6. Find the measure of each angle in the diagram.
z2
400
2z