Today, we will be investigating reflections that occur across two lines that intersect at any angle, that intersect at right angles, and that never intersect within the same plane. In other words, you will be reflecting an object across one line of reflection and then its image across a second line of reflection. You will then compare the pre-image to the second image that is obtained.
Sketch 1
Step 1 : Click here to open an existing GSP sketch from which you will do your work on reflections.
Step 2 : Close in the GSP sketch and this Day 5 set of instructions so that they sit side by side. This will enable you to read the instructions and work on the sketch at the same time.
Step 3 : Reflect quadrilateral GLMK across line AC.
Step 4 : Label the vertices of the image.
Step 5 : Reflect the image G'L'M'K' across line AE.
Step 6 : Label the vertices of the second image.
Step 7 : Measure angle GQG'. (You could just as easily measure angle KQK' or angle LQL' or angle MQM'.)
Step 8 : Highlight the original pre-image (quadrilateral GLMK) and finally point A.
Step 9 : Under the Transform menu, select Mark Center and then Rotate. Fill in the angle of rotation as the angle you found when you measured angle GQG'.
Step 10 : Measure angle CAE, the angle formed by the intersecting lines.
Investigation 1
What happened when you rotated the pre-image by the measure of angle GQG'? What is the relationship of angle GQG' and angle CAE, the angle of rotation? Double click on the sequence button. What happens to angles GQG' and CAE as the sequence unfolds? What happens to the double reflections you made? Make a conjecture about reflections across two lines that intersect.
Sketch 2
Step 1 : Click here to open an existing GSP sketch from which you will do your work on reflections.
Step 2 : Close in the GSP sketch and this Day 5 set of instructions so that they sit side by side. This will enable you to read the instructions and work on the sketch at the same time.
Step 3 : Reflect quadrilateral GFHE across line CD.
Step 4 : Label the vertices of the image.
Step 5 : Reflect the image G'F'H'E' across line AB.
Step 6 : Label the vertices of the second image.
Step 7 : Under the Graph menu, select Show Grid.
Step 8 : After the grid is showing, count the vertical and horizontal moves for a vector to translate the original quadrilateral GFHE to the second image, quadrilateral G"F"H"E".
Step 9 : Select the quadrilateral GFHE, being sure to highlight all the vertices, sides, as well as the interior.
Step 10 : Under the Transform menu, select Translate. Then select By Rectangular Vector. Fill in the vertical and horizontal components you counted in step 8.
Investigation 2
What happened to quadrilateral GFHE when you made the translation? Move one of the lines of reflection and see if your answer holds. Drag one of the points on either line AB or CD. What happens to the pre-image and the images? Does the translated quadrilateral remain the same? Make a conjecture about reflections that occur across parallel lines.
Assignment - Day 5
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.