Geometric Transformations - Reflection
This activity allows you to experiment with the transformation known as reflection.
A reflection over a line, k, is a transformation in which each point of the original figure 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. i.e) it is a "flip" of the image.
Through an investigation involving three parts, you will learn some of the rules associated with reflecting points and figures over various (important) lines of reflection.
Begin by clicking on the box that says "Part 1." A triangle with vertices at A=(3,7), B=(2,4), and C=(10,5) will appear as well as a red line. This red line will be our line of reflection for our first set of trials and is known as y=0. Can you think of another name for this line?
Part 1: Reflection over the x-axis.
Keeping the vertices of triangle ABC at A=(3,7), B=(2,4), and C=(10,5) predict what the coordinates of it's reflection will be and record your prediction on your worksheet.
Reminder: A reflection of a single point will have the same distance from the line of reflection but on the other side of the line.
Check your prediction by clicking on the "Reflected Shape and Coordinates" box- A blue figure should appear with coordinates A', B', and C' labeled, this is your reflected figure.
How do the coordinates of the resulting reflection compare with your predictions?
Can you see anything interesting between the coordinates of the original triangle and its reflection?
When you are done thinking about this you may unclick the "Reflected Shape and Coordinates" box. THIS STEP MUST BE DONE AFTER EACH INDIVIDUAL TRIAL
If you haven't already noticed...the vertices of triangle ABC can be moved!!! And that is exactly what I want you to do for your next trial!
Move one, two or all three of the vertices of triangle ABC to a new position and record the coordinates on your worksheet.
Predict what the coordinates of the reflection will be and check your answer using the "Reflected Shape and Coordinates" box.
What did you find after this trial? Anything interesting?
Run a third trial, again moving the vertices of triangle ABC to a new position on the grid. Record your original coordinates and predict what the coordinates of the reflection will be. Check your answers using the "Reflected Shape and Coordinates" box.
After running all three trials, what can be said about the relationship between the coordinates of the original figure and the coordinates of its reflection? Is there any special rule or trick you can see to know where the coordinates should lie when reflected over the x-axis?
FInish this Part by first unclicking the "Reflected Shape and Coordinates" box and then unclicking the "Part 1" box
Click on "Part 2" and this time a quadrilateral with vertices at D=(6,8), E=(12,6), F=(10,2), and G=(8,4) will appear. The red line is again our line of reflection and is defined by x=0. Can you think of another name for this line?
Part 2: Reflection over the y-axis.
Keeping the vertices of quadrilateral DEFG at D=(6,8), E=(12,6), F=(10,2), and G=(8,4) predict what the coordinates of it's reflection will be and record your prediction on your worksheet.
Check your prediction by clicking on the "Reflected Shape and Coordinates" box.
How do the coordinates of the resulting reflection compare with your predictions?
Can you see anything interesting between the coordinates of the original quadrilateral and its reflection?
Unclick the "Reflected Shape and Coordinates" box and then move any combination of the vertices of quadrilateral DEFG to a new position and record the coordinates on your worksheet.
Predict what the coordinates of the reflection will be and check your answer using the "Reflected Shape and Coordinates" box.
Run a third trial, predicting the output of each vertices when reflected of the y-axis. Check your answers using the "Reflected Shape and Coordinates" box.
After running all three trials, what rule or trick can you see to know where the coordinates should lie when reflected over the y-axis? How does this compare to what you found in Part 1?
When you are finished thinking about this, you may unclick first the "Reflected Shape and Coordinates" box and then the "Part 2" box.
Part 3: For this final trial we will now be moving the line of reflection rather than the vertices of the figure.
Click on the "Part 3" box. In the upper left corner you will see a slider. Move the slider all the way to the right until it reads m=1.
This slider moves the line of reflection into a position that should look familiar to you. Can you identify the equation of our line of reflection?
Set 1: Reflection over the line y=x.
Keeping the vertices of triangle HIJ at H=(10,2), I=(6,4), and J=(10,6) predict where the coordinates of its reflection will lie and record on your worksheet. Check your answers with the "Reflected Shape and Coordinates" box.
Move the vertices of the triangle and run two more trials.
What pattern can you see between the coordinates in this case? Can you see a rule forming? Does this make sense? Why or Why not?
When you are finished with this trial, unclick the "Reflected Shape and Coordinates" box and move the slider now all the way to the left so that m = -1. Can you give the definition of this line of reflection?
Set 2: Reflection over the line y=-x
Keep the vertices of triangle HIJ at H=(10,2), I=(6,4), and J=(10,6) for your first trial. Predict, record and check your findings for the coordinates of the reflected image.
Run two more trials with different placement of vertices of triangle HIJ. Predict, record and check your findings.
What rule applies in this case? How does it compare to the rule when reflecting over the line y=x?