Constructing Perpendiculars Lesson Summary: Students will complete the basic compass and straight edge constructions commonly taught in first year high school Geometry. Key Words: perpendicular, compass, ruler, constructions Background Knowledge: It is assumed that students will have been exposed to and are familiar with the following terms: perpendicular, bisector, midpoint, construction, and algorithm. It is also assumed that students are familiar with the use of a compass and a ruler. The instructor may choose to review these terms prior to beginning the lab. Learning Objectives: 1. Construct perpendicular bisector of a given segment. 2. Construct the perpendicular line from point to a line. 3. Construct the perpendicular to a line at a point. Materials: Computer lab or set of calculators equipped with Cabri Geometry II and lab worksheet. Suggested Procedure: Group students in pairs. The instructor may choose the method to determine pairings. Using the overhead or chalkboard, draw figure 1. Pose the following questions to the students. You are at the park one afternoon when you see someone has fallen while rollerblading on the paths and he has been injured. You need to run to the main road to flag down some help. What is the shortest route to take? The students should be able to determine that they should run from their position along the line perpendicular to the road. Use this to move into the lab on perpendiculars. Figure 1 main road You are here. Have students work in pairs to perform the following constructions.
Constructing Perpendiculars Team Members Names: Goal 1: Construct the perpendicular bisector of a given segment. Investigate Using Cabri Geometry II* 1. Construct segment AB. [Use segment tool] 2. Construct a circle with center A that passes through point B. [Use circle tool] 3. Construct a second circle with center B that passes through point A. 4. What can you conclude about the radius of each circle? 5. Create and label the points where the two circles intersect [Use point and label tools] C and D. 6. Construct segment CD.
7. Find the intersection point of AB and CD. Label this point M. 8. Measure segments AM and MB. 9. What do you observe about these values? 10. Now measure CMA and CMB. 11. What do you observe about the values of these angles? What does this tell you about the relationship between segments AB and CD?
Goal 2: Given a point on a line construct the perpendicular to the line through the point. 12. Construct line m. [use line tool] 13. Create a point A on line m. [use point tool] 14. Create a second point P on line m. 15. Now construct a circle with center A, passing through point P. [use circle tool] 16. This circle intersects line m at a second point. Label this point B. 17. What does point A represent with respect to segment BP? Explain. 18. Could you use the previous approach from goal 1 to create a perpendicular to line m at point A? Explain. Let s see how well you ve mastered these constructions. 19. Construct the perpendicular bisector to the given segment LM. 20. Construct the line perpendicular to line l through point P.
Extension 21. Given a circle with center O, passing through points A and B, describe how to construct two perpendicular diameters. 22. Given ray AB, describe how to construct right angle CAB.
Journal Activity Constructing Perpendiculars 1. What was your favorite thing about this activity? 2. What was the most challenging thing? 3. What did you gain the most confidence about through completing this lesson? 4. Where do you possibly see yourself using this knowledge in the future?