Introduction
A line of symmetry, , is a line separating a figure into
two halves that are mirror images. Line symmetry
exists for a figure if for every point P on one side of the
line, there is a corresponding point Q where is the
perpendicular bisector of
.
1
5.1.3: Applying Lines of Symmetry
Introduction, continued
From the diagram, we see that is perpendicular to
.
The tick marks on the segment from P to R and from R
to Q show us that the lengths are equal; therefore, R is
the point that is halfway between
.
Depending on the characteristics of a figure, a figure
may contain many lines of symmetry or none at all. In
this lesson, we will discuss the rotations and reflections
that can be applied to squares, rectangles,
parallelograms, trapezoids, and other regular polygons
that carry the figure onto itself. Regular polygons are
two-dimensional figures with all sides and all angles
congruent.
5.1.3: Applying Lines of Symmetry
2
Introduction, continued
Squares
Because squares have four equal sides and four equal
angles, squares have four lines of symmetry. If we rotate
a square about its center 90˚, we find that though the
points have moved, the square is still covering the same
space.
Similarly, we can rotate a square 180˚, 270˚, or any other
multiple of 90˚ with the same result.
3
5.1.3: Applying Lines of Symmetry
Introduction, continued
We can also reflect the square through any of the four
lines of symmetry and the image will project onto its
preimage.
Rectangles
A rectangle has two lines of symmetry: one vertical and
one horizontal. Unlike a square, a rectangle does not
have diagonal lines of symmetry. If a rectangle is rotated
90˚, will the image be projected onto its preimage? What
if it is rotated 180˚?
4
5.1.3: Applying Lines of Symmetry
Introduction, continued
5
5.1.3: Applying Lines of Symmetry
Introduction, continued
If a rectangle is reflected through its horizontal or vertical
lines of symmetry, the image is projected onto its
preimage.
Vertical reflection
Horizontal reflection
A,
B'
B,
A'
A, D'
B, C'
D, C'
C, D'
D, A'
C, B'
6
5.1.3: Applying Lines of Symmetry
Introduction, continued
Trapezoids
A trapezoid has one line of symmetry bisecting, or
cutting, the parallel sides in half if and only if the nonparallel sides are of equal length (called an isosceles
trapezoid). We can reflect the isosceles trapezoid shown
below through the line of symmetry; doing so projects
the image onto its preimage. However, notice that in the
last trapezoid shown on the next slide,
is longer than
, so there is no symmetry. The only rotation that will
carry a trapezoid that is not isosceles onto itself is 360˚.
7
5.1.3: Applying Lines of Symmetry
Introduction, continued
8
5.1.3: Applying Lines of Symmetry
Introduction, continued
Parallelograms
There are no lines of symmetry in a parallelogram if a
90˚ angle is not present in the figure. Therefore, there is
no reflection that will carry a parallelogram onto itself.
However, what if it is rotated 180˚?
9
5.1.3: Applying Lines of Symmetry
Introduction, continued
10
5.1.3: Applying Lines of Symmetry
Key Concepts
• Figures can be reflected through lines of symmetry
onto themselves.
• Lines of symmetry determine the amount of rotation
required to carry them onto themselves.
• Not all figures are symmetrical.
• Regular polygons have sides of equal length and
angles of equal measure. There are n number of lines
of symmetry for a number of sides, n, in a regular
polygon.
11
5.1.3: Applying Lines of Symmetry
Common Errors/Misconceptions
• showing a line of symmetry in a parallelogram or
rhombus where there isn’t one
• missing a line of symmetry
12
5.1.3: Applying Lines of Symmetry
Guided Practice
Example 1
Given a regular pentagon ABCDE, draw the lines of
symmetry.
13
5.1.3: Applying Lines of Symmetry
Guided Practice: Example 1, continued
1. First, draw the pentagon and label the
vertices.
Note the line of symmetry from A to
.
14
5.1.3: Applying Lines of Symmetry
Guided Practice: Example 1, continued
2. Now move to the next vertex, B, and
extend a line to the midpoint of
.
15
5.1.3: Applying Lines of Symmetry
Guided Practice: Example 1, continued
3. Continue around to each vertex,
extending a line from the vertex to the
midpoint of the opposing line segment.
Note that a regular pentagon has five sides,
five vertices, and five lines of reflection.
✔
16
5.1.3: Applying Lines of Symmetry
Guided Practice: Example 1, continued
17
5.1.3: Applying Lines of Symmetry
Guided Practice
Example 3
Given the quadrilateral ABCE, the square ABCD, and
the information that F is the same distance from A and
C, show that ABCE is symmetrical along
.
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5.1.3: Applying Lines of Symmetry
Guided Practice: Example 3, continued
1. Recall the definition of line symmetry.
Line symmetry exists for a figure if for every point on
one side of the line of symmetry, there is a
corresponding point the same distance from the line.
We are given that ABCD is square, so we know
.
@
We also know that
.
We know
@
.
5.1.3: Applying Lines of Symmetry
is symmetrical along
19
Guided Practice: Example 3, continued
@
2. Since
and
,
is a line
@
of symmetry for
where
.
20
5.1.3: Applying Lines of Symmetry
Guided Practice: Example 3, continued
3.
has the same area as
because they share a base and have
equal height.
.
@ , so
21
5.1.3: Applying Lines of Symmetry
Guided Practice: Example 3, continued
4. We now know
is a line of symmetry
for
and
is a line of symmetry for
, so
and
quadrilateral ABCE is symmetrical along
.
✔
22
5.1.3: Applying Lines of Symmetry
Guided Practice: Example 3, continued
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5.1.3: Applying Lines of Symmetry