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 ReflectionsDraw 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

D

BC

B’

D’

A’

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 ReflectionsNow 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 ReviewThe 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 TranslationsDraw 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

D

B

C

D’

B’

A’

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 TranslationsNow 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 RotationsDraw 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 RotationsUse 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.