Download line symmetry of a figure - Manhasset Public Schools

CC Geometry R
Aim #18: What types of symmetry does a figure have?
Do Now: In the diagram below:
Sketch Triangle B, the reflection of Triangle A across line x.
Sketch Triangle C, the reflection of Triangle B across line y.
Complete: Line x and line y are each a _______ __ ________________.
y
A student did this also, looked at Triangle C and said,
"That's not the image of triangle A after two
reflections - that's the image of triangle A
after a __________________!!"
(He was right!)
A
X
When a figure is reflected across a line, and
the resulting image is reflected across a second
line that intersects the first line, the final image
is a ______________ of the original figure.
A figure has symmetry if there exists a rigid motion (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.
Exercise 1: Draw in all lines of symmetry for each object.
C
parallelogram
rectangle
equilateral Δ
square
isosceles Δ
Exercise 2: Construct a line of symmetry for isosceles ΔABC. (AB = CB)
B
A
C
Regular Polygon: A polygon is regular if all sides have equal length (is
equilateral) and all angles have equal measure (is equiangular).
Exercise 3: Construct one line of symmetry that does not intersect a vertex.
Then sketch in all remaining lines of symmetry.
regular hexagon
ROTATIONAL AND IDENTITY SYMMETRY OF A FIGURE
A figure has rotational symmetry if the figure can be mapped onto itself by a
rotation (turn) between 0˚and 360˚ about the center of the figure, called the
center of rotation.
Exercise 4: Determine the figures below that have rotational symmetry, and
state all the angles that produce rotational symmetry.
square
rectangle
regular pentagon
right triangle
Exercise 5:Construct and label D, the center of rotation for each regular polygon.
Exercise 6: In the equilateral triangle to the right,
0
with circumcenter D, ______ rotations of _____
Exercise 7: ABCD is a square.
D
A
B
D
C
a) Draw all lines of symmetry.
Label the center of rotation, S.
b) What kinds of symmetry does the square have?
c) Name all angles of rotational symmetry.
Exercise 8: The figure is a regular octagon.
a) How many lines of symmetry are there?
b) Name all the angles of rotational symmetry.
Exercise 9: What is the minimum number of degrees that maps
a) a regular hexagon onto itself?
b) a regular octagon onto itself?
c) a regular polygon with 10
d) a regular polygon with n
sides onto itself?
sides onto itself?
Exercise 10:
James says that the figure has only line symmetry.
Jewel says that the figure has only rotational symmetry.
Is either of them correct? Explain.
Exercise 11: Draw a figure that has rotational symmetry, but not line symmetry,
not yet in this Aim.
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
0
0
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.