E9 Students are expected to make
generalizations about the properties
of translations and reflections and
apply these properties.
E10
Students are expected to
explore rotations of one-quarter,
one-half, and three-quarter turns
using a variety of centres.
What is Transformational Geometry?
Transformational geometry is the study of
transformations.
What is a transformation?
A transformation is a change in size or position of
a geometric figure.
Types of Transformations
Reflections or what are called ‘flips’ in the
younger grades.
Translations or what are called ‘slides’ in
the younger grades.
Rotations or what are called ‘turns’ in the
earlier grades.
Dilatations that are introduced in Grade
Six.
What is a Reflection in
Transformational Geometry?
A reflection is the figure formed by flipping
or reflecting a geometric figure about a line
to get a mirror or reflection image.
The original figure is referred to as the
object while the figure created by the
reflection of that object is called the
reflection image.
Now let’s create some reflection images
and try to discover the properties of a
reflection.
Investigating Reflections
Draw a square on white paper. Draw a mirror
line or a line of reflection. Place your mira on this
line, and then draw the reflected image.
Draw another shape. Draw a mirror line or a line
of reflection. Place your mira on this line, and
then draw the reflected image.
Compare your original drawings (objects) with
the reflected images by tracing the object and
then fitting it on its reflected image. What can
you say about the object and its reflected
image?
You should have concluded that the object and
its reflected image are congruent.
Investigating Reflections
Draw a trapezoid and label its vertices with
A, B, C, and D.
Next, label the corresponding vertices of
the reflected image with A’, B’, C’, D’.
Name both shapes clockwise starting at A
and A’. What do you notice?
You should have noticed that the object
and its image are of opposite orientation.
See the next slide to review this.
What do you notice
in the diagram below?
A
A’
D’
D
B
C
B’
C’
The object and its reflected image are of opposite orientation.
Investigating Reflections
Now examine the trapezoid and its reflected
image again.
Make
line
segments
by
joining
the
corresponding vertices (A to A’, etc.). Examine
the angles made by the mirror lines with these
segments. What can you conclude?
You should have concluded that the mirror lines
are perpendicular to the line segments joining
the corresponding image points.
Investigating Reflections
Now examine the trapezoid and its reflected
image again.
Measure the distance from one vertex in the
original trapezoid to the mirror line. Then
measure the distance of the corresponding
vertex in its image to the mirror line. Do this for
the other pairs of points. What do you notice?
You should have noticed that the corresponding
points are equidistant from the mirror lines.
You should now notice that the mirror line is the
perpendicular bisector of all segments joining
corresponding points.
Properties of Reflections: A Review
The image is congruent to the object.
The orientation of the image is reversed. If triangle
ABC is read clockwise, then triangle A’B’C’ is read
counterclockwise.
If corresponding points on the object and image are
joined, each line segment makes a right angle with
the reflection or mirror line. (perpendicular). In other
words, the mirror lines are perpendicular to the line
segments joining the corresponding image points.
Any point and its image are the same distance from
the reflection line. The mirror line is the
perpendicular bisector of all segments joining
corresponding points.
Let’s Practice!
Use the properties of reflections to draw
reflected images from objects presented to
you by the teacher.
What is a Translation in
Transformational Geometry?
A translation is a movement of a geometric
figure to a new position by sliding it along
a plane.
The original figure is referred to as the
object while the figure created by the
translation or slide of that object is called
the translated image or sometimes just the
image.
Now let’s create some translated images
and try to discover the properties of a
translation.
Investigating Translations
Draw a square on white paper. Trace
around the square and then take this
tracing and translate it on the plane or grid
paper according to the slide rule given to
you by the teacher.
Compare the size of the object with its
translated image. What can you say about
the object and its image?
You should have concluded that the object
and its translated image are congruent.
Investigating Translations
Draw a trapezoid and label its vertices with
A, B, C, and D.
Next, label the corresponding vertices of
the translated image with A’, B’, C’, D’.
Name both shapes clockwise starting at A
and A’. What do you notice?
You should have noticed that the shape
and its image are of the same orientation.
See the next slide to review this.
What do you notice
in the diagram below?
A
A’
D’
B’
D
C’
B
C
The object and its translated image are of the same
orientation.
Investigating Translations
Now examine the trapezoid and its translated
image again.
What do you notice about line segment AB and
its corresponding line segment A’B’? What about
line segment DC and line segment D’C’? Look at
the other corresponding line segments to see if
you notice the same property.
You should have noticed that the corresponding
sides of the object and its translated image are
parallel to one another.
Investigating Translations
Now examine the trapezoid and its translated
image again.
Measure the distance from one vertex in the
original trapezoid to its corresponding vertex in
the image. Next, measure the distance from a
second vertex in the object to its corresponding
vertex in the image. Do the same for the two
other sets of corresponding vertices. What do
you notice about these measurements?
You should have noticed that all of the line
segments joining corresponding points are equal
in length and parallel to one another.
Properties of Translations: A Review
The image is congruent to the object.
The image has the same orientation as the
object.
The line segments of the image are parallel to
the corresponding line segments of the object.
The line segment joining the vertices of the
object to the corresponding image vertices are
equal in length and parallel.
Let’s Practice!
Use the properties of translations to draw
translated images from objects presented
to you by the teacher.
What is a Rotation in
Transformational Geometry?
A rotation is the movement that results when a
geometric figure is turned about a fixed point.
The original figure is referred to as the object
while the figure created by the reflection of that
object is called the rotated image.
A rotation can be created by rotating an object
about its centre point or about a another fixed
point on the figure even a point somewhere
outside of the figure on the plane.
Investigating Quarter-Turn Rotations
Draw a square on white paper. Mark its vertices with
the points A, B, C, and D. Next, mark the square’s
centre point with the point E. Hold the square at this
centre point, and rotate it 90 degrees.
This is called a quarter turn of the square about its
center point. Save your square for a later activity.
Now use a different figure such as a trapezoid. Label
its vertices A, B, C, and D and its centre point E.
Now make a quarter turn about the trapezoid’s
centre. Compare the object with its rotated image.
What can you say about the size of the object
compared to its image?
You should have concluded that the object and its
rotated image are congruent. Save your trapezoid.
Investigating Half-Turn Rotations
Use your labeled square from the previous set of
activities. Hold the square at this centre point, and
this time rotate it 180 degrees.
This is called a half turn of the square about its
center point.
Now use your trapezoid from the previous set of
activities. This time use it to make a half turn about
the trapezoid’s centre. Compare the object with its
rotated image. What can you say about the size of
the object compared to its image?
You should have concluded that the object and its
rotated image are congruent.
Investigating Quarter-Turn Rotations
Examine your object (trapezoid) and its
image again.
What labels should you place on the
rotated image?
Name both shapes clockwise starting at A
and A’. What do you notice?
You should have noticed that the shape
and its rotated image are of the same
orientation.
Investigating Half-Turn Rotations
Examine your object (trapezoid) and its
image again.
What labels should you place on the
rotated image?
Name both shapes clockwise starting at A
and A’. What do you notice?
You should have noticed that the shape
and its rotated image are of the same
orientation.
Investigating Quarter-Turn Rotations
Examine your object (trapezoid) and its image
again.
Measure the distance between any point on the
object to the centre of the object. Now measure
the distance between that point on the image to
the centre of the image. What do you notice?
You should have noticed that any point and its
image are the same distance from the rotation
centre.
Investigating Half-Turn Rotations
Examine your object (trapezoid) and its image
again.
Measure the distance between any point on the
object to the centre of the object. Now measure
the distance between that point on the image to
the centre of the image. What do you notice?
You should have noticed that any point and its
image are the same distance from the rotation
centre.
Investigating Quarter-Turn Rotations
Examine your object (trapezoid) and its image
again.
Place a tracing of the image on top of the object.
Draw a line segment from any point on the object
to the turn centre of the object. Draw a similar line
segment from a corresponding point on the image
to the centre of the image. What do you notice
about the angle formed by these two segments?
You should have noticed corresponding points
drawn to the turn centre will form the angle of turn,
in this case a 90-degree angle.
Investigating Half-Turn Rotations
Examine your object (trapezoid) and its image
again.
Place a tracing of the image on top of the object.
Draw a line segment from any point on the object
to the turn centre of the object. Draw a similar line
segment from a corresponding point on the image
to the centre of the image. What do you notice
about the angle formed by these two segments?
You should have noticed corresponding points
drawn to the turn centre will form the angle of turn,
in this case a 180-degree angle.
Investigating Half-Turn Rotations
Now examine the trapezoid and its rotated
image again.
What do you notice about line segment AB and
its corresponding line segment A’B’? What about
line segment DC and line segment D’C’? Look at
the other corresponding line segments to see if
you notice the same property.
You should have noticed that the corresponding
sides of the object and its half-turn rotated image
are parallel to one another.
Properties of Quarter-Turn
Rotations: A Review
The image is congruent to the object.
The image has the same orientation as the
object.
Any point and its image are the same distance
from the rotation centre.
Corresponding points drawn to the turn centre
will form the angle of turn.
Properties of Half-Turn Rotations:
A Review
The image is congruent to the object.
The image has the same orientation as the
object.
Any point and its image are the same distance
from the rotation centre.
Corresponding points drawn to the turn centre
will form the angle of turn.
Pairs of corresponding line segments are
parallel.
Investigating More about Rotations
Now make three-quarter rotations of one
of your figures. What properties do you
notice?
Now investigate with the teacher to see if
the properties of quarter, half, and three
quarter rotations hold turn if the turn point
is not the centre of the object but another
point on the object or a point that is not
located on the object but is somewhere
outside of it.
Well, students, this is about all I
can tell you right now about
reflections,
translations,
and
rotations. Stay tuned for Grade Six.