[b]Reflections in geometry have some of the same properties of a reflection you observe when looking into a mirror. In this activity, you’ll investigate the properties of reflections that make a reflection the “mirror image” of the original. [/b][br]
1. Observe ΔCDE in the sketch window below. [i]Predict[/i] where the image of Δ[i]CDE[/i] will be once reflected over line AB. Construct points for vertices C’, D’ and E’ in your sketch. [[u]Technology Tip[/u]: Along the top of the sketch window, you will see a row of buttons, referred to as the [i]Tool Bar[/i]. Under the second button from the left, select [b]Point[/b]; and click anywhere in your sketch window to create a point.]
2. Reflect Δ[i]CDE[/i] (both its sides and vertices) over line [i]AB[/i]. [br][br][[u]Technology Tip[/u]: Along the tool bar, under the third button from the right, select [b]Reflect Object about Line[/b]; select the triangle CDE and the line of reflection.]
3. Assess your prediction points. How close were [i]your prediction points[/i] to the (actual) image vertices of Δ[i]CDE[/i]?[br][br][br]
4. [i]Predict [/i]the movement of the triangles when you [u]drag the vertices of the original [/u][u]ΔCDE[/u]. Observe and describe how the triangles are related. Also, be sure to drag the line of reflection.
5. Are a figure and its mirror image always congruent under manipulation of the triangle or of the mirror? Why?
6. Construct segments connecting each point and its image. Make these segments dashed. [br][br][[u]Technology Tip[/u]: Under the third button from the left along the top tool bar, select [b]Segment[/b]; then select corresponding vertices on the triangles. Remember to go back to the [b]Selection Arrow[/b] (the first button on the left) to end drawing segments. Click on the segment in your sketch, and look for a button on the top right corner of your sketch window with three segments and a purple triangle and circle. Click on that button, use the options on the [b]Color[/b] and [b]Style[/b] sub-buttons.]
7. Predict the relationships between the dashed segments and the mirror line. Drag the vertices and sides of the triangle around and observe the relationship.
8. [Challenge] Suppose GeoGebra didn’t have a [b]Reflection Tool[/b]. How could you construct a given point’s[br]reflected image over a given line? Try it. Start with a point and a line (as shown in the sketch below). Come up with a construction for the reflection of the point over the line. Describe your method using both words and sketches.
In this part of the exploration, you’ll investigate what happens to the coordinates of points when you reflect them across the [i]x[/i]- and [i]y[/i]- axes in the coordinate plane.
1. In the sketch window below, draw Δ[i]ABC[/i] with vertices on the grid. [[u]Technology Tip[/u]: You can use either the point tool and connect the points by segments; or you can use the segment tool to construct both your vertices and sides of your triangle.]
2. Record the coordinates of each of the vertices below.
3. Make a prediction where the image of Δ[i]ABC[/i] would lie once reflected over the [i]y[/i]-axis. Then reflect Δ[i]ABC[/i] over the [i]y[/i]-axis to assess your prediction. Record your results below.[br][i][br]Reflected over the y-axis [b]prediction[/b][/i]: A′ = ______ B′ = ______ C′ = ______[br][br][i]Reflected over the y-axis [b]actual[/b][/i]: A′ = ______ B′ =______ C′ =[b] _______[/b]
Drag the vertices to different points on the grid and look for a relationship between the point’s coordinates and the coordinates of its image when reflected across the [i]y[/i]-axis.
4. Describe any relationship you observe between the coordinates of the vertices of your original triangle and the coordinates of the reflected image across the [i]y[/i]-axis.
5. Draw a new triangle in the sketch window below. Record the coordinates of each of the vertices below.
6. Make a prediction where the image of Δ[i]ABC[/i] would lie once reflected over the [i]x[/i]-axis. Then reflect Δ[i]ABC[/i] over the [i]x[/i]-axis to assess your prediction. Record your results below.[br][i][br]Reflected over the x-axis [b]prediction[/b][/i]: A′ = ______ B′ = ______ C′ = ______[br][br][i]Reflected over the x-axis [b]actual[/b][/i]: A′ = ______ B′ =______ C′ =[b] _______[/b]
Drag the vertices to different points on the grid and look for a relationship between the point’s coordinates and the coordinates of its image when reflected across the [i]x[/i]-axis.
7. Describe any relationship you observe between the coordinates of the vertices of your original triangle and the coordinates of the reflected image across the [i]x[/i]-axis. Delete your triangle’s image.
8. Graph the line [i]y = x[/i], by typing in the equation in the [b]Input line[/b]. Reflect your triangle across this line. Describe any relationship you observe between the coordinates of the original points and coordinates of their reflected images across the line [i]y = x[/i]. [br][br][i]Reflected over the line y = x [/i][b]actual[/b]: A′= ______ B′ = ______ C′ = _______
9. Make some overall general statements about reflections. What were some of the main results from today's tasks.