In this activity, you will investigate what happens to the coordinates of point when they are reflected across the x-axis, the y-axis, and across the line y=x in the coordinate plane.
Sketch 1
Step 1 : Open a new sketch in Geometer's Sketchpad. Under the Graph menu, select Show Axis and Show Grid.
Step2 : Use the Line Tool icon, and create a line on the x-axis. Under Display, select Color and change the color of the line you have created so that it is more visible.
Step 3 : Create the following triangle using these points: A(-3,2), B(-2,3), and C(-2,0). You will notice that this triangle is a scalene and obtuse. After highlighting the three vertices, under Construct, select Polygon Interior to shade in the triangle. You may want to also go to Display and select Color to change the color of the triangle.
Step 4 : Select the triangle as well as its vertices and sides, and finally select the x-axis using the Select Tool icon.
Step 5 : Under Transform, select Mark Mirror. Then go back to the Transform menu and select Reflect.
Step 6 : Using the Label Tool, label the vertices of the image.
Investigation 1
What are the coordinates of A', B', and C'? How do the coordinates of the image relate to those corresponding points of the pre-image? Go back to your sketch and create a new object to be reflected across the x-axis. How do the the coordinates of the pre-image relate to their corresponding coordinates in the image? What conjecture can you make about the coordinates of objects that are reflected across the x-axis?
Sketch 2
Step 1 : Go back to the GSP sketch you used in Sketch 1.
Step 2 : Delete the line you created on the x-axis.
Step 3 : Create a line on the y-axis this time. Change the color of this line to that it is easier to see. (Use Step 2 in Sketch 1 to accomplish this.)
Step 4 : Select the triangle as well as its vertices and sides, and finally select the y-axis using the Select Tool icon.
Step 5 : Under Transform, select Mark Mirror. Then go back to the Transform menu and select Reflect.
Step 6 : Using the Label Tool, label the vertices of the image.
Investigation 2
What are the coordinates of A', B', and C'? How do the coordinates of the image relate to those corresponding points of the pre-image? Go back to your sketch and create a new object to be reflected across the y-axis. How do the the coordinates of the pre-image relate to their corresponding coordinates in the image? What conjecture can you make about the coordinates of objects that are reflected across the y-axis?
Sketch 3
Step 1 : Go back to the GSP sketch you used in Sketch 2.
Step 2 : Delete the line you created on the y-axis.
Step 3 : Create a line by using the Line Tool icon. Begin the line by starting at (0,0) and be sure it goes through (1,1). This is the line y=x. (You will notice that in the given ordered pairs, the x-coordinate and the y-coordinate in each case are equal.) You will need to color this line a different color as you did in the first two sketches.
Step 4 : Select the triangle as well as its vertices and sides, and finally select the line y=x using the Select Tool icon.
Step 5 : Under Transform, select Mark Mirror. Then go back to the Transform menu and select Reflect.
Step 6 : Using the Label Tool, label the vertices of the image.
Investigation 3
What are the coordinates of A', B', and C'? How do the coordinates of the image relate to those corresponding points of the pre-image? Go back to your sketch and create a new object to be reflected across the line y=x. How do the the coordinates of the pre-image relate to their corresponding coordinates in the image? What conjecture can you make about the coordinates of objects that are reflected across the line y=x?
Assignment - Day 4
1. Answer the questions posed in the investigations above.
2. Describe the new ideas you learned about today.
3. Discuss any questions or points of misunderstanding you may have had in the course of this lesson.
4. Use the conjectures you made in the sketches and investigations to answer each of the following-
a. Write the coordinates of the following points as if they were reflected across the x-axis without using a grid:
i. (2,4)
ii. (-5,-16)
iii. (-6,7)
iv. (3,-5)
v. (0,-8)
vi. (3,0)
b. Write the coordinates of the points given above as if they were reflected across the y-axis without using a grid.
c. Write the coordinates of the points given above as if they were reflected across the line y=x without using a grid.