Please watch PPT: line symmetry then complete.
Transformational Geometry
Lesson 10-1 Symmetry and reflection
KEY
LEADER: Line symmetry, or just symmetry, occurs when two halves of a figure mirror each
other across a line. The line of symmetry is the line that divides the figure into two mirror
images. A simple test to determine if a figure has line symmetry is to fold the figure along the
supposed line of symmetry and see if the two halves of the figure coincide.
Plane symmetry occurs when a plane intersects a three dimensional figure such that one half is the
reflected image of the other half. Such a plane is imaginary and divides an object into two halves,
each of which is the mirror image of the other in this plane.
Plane symmetry is analogous to line symmetry, only in three dimensions. Common objects
displaying plane symmetry are rectangular solids, spheres, boxes, cones, and humans.
Examples of Symmetry
Vertical line
Horizontal line
Both:
All:
1. If the alphabet were printed in simple block printing, which capital letters would have
vertical line symmetry?
A H I M O T U V WX Y
2.
If the alphabet were printed in simple block printing, which capital letters would have
horizontal line symmetry?
B C D E HIKO X
3. If the alphabet were printed in simple block printing, which capital letters would have BOTH
vertical and horizontal symmetry?
H I O X
Please copy yellow highlighted vocab.
General "Transformation" Vocabulary
Transformational Geometry: is a method for studying geometry that illustrates
congruence and similarity by the use of transformations.
Transformation: A transformation of the plane is a one-to-one mapping of points in the
plane to points in the plane.
 Reflection- is a transformation in which each point of the original figure (preimage) has an image that is the same distance from the line of reflection as the
original point but is on the opposite side of the line.
 Rotation- is a transformation that turns a figure about a fixed point called the
center of rotation.

Translation- is a transformation that "slides" an object a fixed distance in a given
direction. A translation creates a figure that is congruent with the original figure.

Dilation- is a transformation that produces an image that is the same shape as
the original, but is a different size. A dilation stretches or shrinks the original
figure.
Opposite Transformation: An opposite transformation is a transformation that
changes the orientation of a figure. Reflections and glide reflections are opposite
transformations.
Image: An image is the resulting point or set of points under a transformation.
Pre-image: original figure or set of points
Isometry: An isometry is a transformation of the plane that preserves length .

Direct: preserves orientation or order - the letters on the diagram go in the same
clockwise or counterclockwise direction on the figure and its image.

Opposite: changes the order (such as clockwise changes to counterclockwise).
Orientation: Orientation refers to the arrangement of points, relative to one another,
after a transformation has occurred. For example, the reference made to the direction
traversed (clockwise or counterclockwise) when traveling around a geometric figure.
Vector: A quantity that has both magnitude and direction; represented geometrically by a
directed line segment.
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Transformational Geometry
Lesson 10-1 Symmetry and reflection
Reflections
Reflection over a line k (notation rk) is a transformation in which each point of the original figure
(pre-image) has an image that is the same distance from the line of reflection as the original
point but is on the opposite side of the line. Remember that a reflection is a flip.
Under a reflection, the figure does not change size.
A line reflection creates a figure that is:
1) congruent to the original figure
2) called an isometry
3) Since naming (lettering) the figure in a reflection requires changing the order of the
letters (such as from clockwise to counterclockwise), a reflection is more specifically called
a __________________________________.
Properties preserved (invariant) under a line
reflection:
1.
2.
3.
4.
5.
-----------------------------------------------------------------6.
Here we see line l and point P not on line l. The reflection of point P in this line will be
point P'. This is stated by rl(P)=P'. The line l will be the perpendicular bisector of the
segment joining point P to point P'.
Reflections in the Coordinate Plane
Reflecting over the x-axis:
(the x-axis as the line of reflection)
When you reflect a point across the x-axis, the x-coordinate__________,
but the y-coordinate is transformed into ___________________.
The reflection of the point (x,
y) across the x-axis is the point (_, __).
Notation: P( x, y)  P ( x, y)
'
or rxaxis ( x, y)  ( x, y)
rx  axis ( FUN ) where
F (-4,2) U (0,2) N( 5,6)
Hint:
rx  axis (FUN )
Answer:
F ' (-4,-2) U' (0,-2) N'( 5,-6)
If you forget the rules
for reflections when graphing,
simply fold your graph paper
along the line of reflection (in
this example the x-axis) to see
where your new figure will be
located. Or you can measure
how far your points are away
from the line of reflection to
locate your new image. Such
processes will allow you to see
what is happening to the
coordinates and help you
remember the rule.
Reflecting over the y-axis:
(the y-axis as the line of reflection)
When you reflect a point across the y-axis, the y-coordinate_______________,
but the x-coordinate is transformed into_____________________.
The reflection of the point (x,
Notation:
y) across the y-axis is the point (-x, y)
or
ry  axis ( FUN ) where
F (-4,2) U (0,2) N( 5,6)
Answer:
ry  axis (FUN ) 
F ' (4,2) U' (0,2) N'( - 5,6)
Reflecting over the line y = x or y = -x:
(the lines y = x or y = -x as the lines of reflection)
When you reflect a point across the line y = x, the x-coordinate and the y-coordinate
change places. When you reflect a point across the line y = -x, the x-coordinate and the ycoordinate change places and are negated (the signs are changed).
The reflection of the point (x,
y) across the line y = x
is the point (__,__).
Write notation: _________________ or ________________________
The reflection of the point (x,
y) across the line y = -x
is the point (__,
__).
Write notation: _________________ or ________________________
ry  x ( FUN ) where
F (-4,2) U (0,2) N( 5,6)
ry  x (FUN )
F ' (2,-4) U' (2,0) N' ( 6,5)
ry   x ( FUN ) where
F (-4,2) U (0,2) N( 5,6)
ry   x (FUN )
F ' (-2,4) U' (-2,0) N' ( - 6,-5)
Reflecting over any line:

Each point of a reflected image is the same
distance from the line of reflection as the
corresponding point of the original figure.

In other words, the line of reflection lies directly in the middle between the
figure and its image -- it is the perpendicular bisector of the segment joining
any point to its image. Keep this idea in mind when working with lines of
reflections that are neither the x-axis nor the y-axis.
COMPLETE THESE GRAPHICALLY!!!
Example #1:
rx  3 (2,6)  _______
Example #2:
ry  4 ( 4,1)  _______
Notice how each point of the original figure and its image are
the same distance away from the line of reflection (which can
be easily counted in this diagram since the line of reflection is
vertical).
ANSWERS:
rx  3 (2,6)  (8,6)
ry  4 (4,1)  (4,9)
Now go try the practice and then check with the key!!!!
Transformational Geometry
Lesson 10-1 Symmetry and reflection
KEY
Reflections
1) The image of the point (2,-9) under a reflection across the x-axis is
2) The image of the point (8,0) under a reflection across the y-axis is
(2, 9)
(-8, 0).
3) The image of the point (-1,-5) under a reflection across the line y = x is
(-5, -1).
4) The triangle is reflected in line l.
Find x, y, z.
2x  1  5
2 y  2  10
2x  4
x2
2 y  12
y6
2z  4
z2
5)
Reflection over the y-axis!
6) Describe what would occur if (-2,9) was reflected in the line y = -x.
(-9, 2)
7) Triangle ABC has coordinates A(-3,3),B(3,2) and C(-1,-4). If the triangle is reflected
over the y-axis, what are the coordinates of image triangle A'B'C'?
A’=(3, 3) B’= (-3, 2) C’= (1, -4)
8) Graph the line of reflection in this diagram and write its equation.
Y= -1
9)
Length and opposite
Name:_____________________________
Geometry Lesson 10-1 Symmetry and Reflection
1) Which letter has both horizontal and vertical line symmetry?
1.
2.
3.
4.
A
S
Z
X
2) Find A', the image of A(3, 5), after a reflection in the line y = -x.
1.
2.
3.
4.
(5, 3)
(-5, -3)
(3, -5)
(-3, 5)
3) The image of point (3, 4) when reflected in the y-axis is
1.
2.
3.
4.
(-3, -4)
(-3, 4)
(3, -4)
(4, 3)
4) What are the coordinates of R', the image of R(-4, 3) after a reflection in the line whose
equation is y = x?
1.
2.
3.
4.
(-4, -3)
(3, -4)
(4, 3)
(-3, 4)
5) Which geometric figure has one and only one line of symmetry?
1.
2.
6) The following graph represents which type of
isometry?
1. Direct
2. Opposite
3. No isometry
3.
4.
7) Which diagram shows a dotted line that is not a line of symmetry?
1.
3.
2.
4.
8) Triangle ABC has vertices A(1,4), B(2,7), C(5,4). Its image is triangle A'B'C' with
vertices A'(1,0), B'(2,-3), C"(5,0).
Graph the triangles and draw the line of
reflection.
Write the equation of the line of reflection.
9)
10) The coordinates of the vertices of quadrilateral ABCD are the points A(1,2),
B(6,1), C(7,6), and D(3,7).
a ) What are the coordinates of quadrilateral A′B′C′D′, the reflection of
quadrilateral ABCD in
the y-axis?
b ) Determine the area, in square
units, of quadrilateral A′B′C′D′.
area = ________square units
11)
2
a) Graph: y  ( x  5) .
b) Graph its reflection in
y=x. Label the graph.
Show the table for y  ( x  5)2
Show the table for the reflection of
y  ( x  5)2
Study your vocabulary now!!!!!! Do not procrastinate!