Project 6.2: Exploring Parallel and Perpendicular Lines
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1 Project 6.2: Exploring Parallel and Perpendicular Lines Name: Date: The project provides an opportunity for students to explore various aspects of Parallel and Perpendicular Lines. Project Scoring Section Reference Max Score Section 1 Parallel Lines 20 Section 2 Perpendicular Lines 20 Creativeness & Neatness (use of color, us of different lines, notations, additional facts, etc.) 10 Scored out of
2 Section 1: Parallel Lines 1. On a xy-plane, graph lines l 1, l 2, and l 3, containing the given points. l 1 contains points A (0,7) and B (8,9); l 2 contains points C (0,4) and D (8,6); l 3 contains points E (0,0) and F (8,2). Make sure to carefully extend the lines past the given points. a. Find the distance between points A and C and between points B and D. What do you notice? What word describes lines l 1 and l 2? b. Find the distance between points C and E and between points D and F. What do you notice? What word describes lines l 2 and l 3? c. Find the distance between points A and E and between points B and F. What do you notice? What word describes lines l 1 and l 3? d. Now find the slopes of l 1, l 2, and l 3. What do you notice? 2
3 2. Now plot line l 4 through points W ( 1,3) and X ( 3,6) and line l 5 through points Y ( 2,1) and Z ( 4,4) carefully extending the lines across the y-axis. a. Use a ruler to measure the distance from W vertically to l 5. Then measure the distance from X vertically to l 5. What do you notice? b. What word describes these lines? c. Find the slope of each line. What do you notice? 3. What appears to be true about the slopes of parallel lines? 3
4 4. Follow the steps below to prove this true for all pairs of parallel lines. a. Let the straight lines l and m be parallel. Sketch these on grid paper. b. Plot any points U and V on line l. Next, draw point W so that WV is the rise and UW is the run of the slope of line l. (A straight line can have only one slope.) That is, slope of line l is WV UW. c. Draw the straight line UW so that it intersects line m at point X and extends to include Point Z such that segment YZ is perpendicular to UW. What is the slope of line m? d. Line UZ is the of the lines l and m, so VUW and YXZ are angles, so VUW YXZ. e. Why is it true that, UWV YXZ? f. Now, Δ UWV and Δ XZY are similar, so the ratio of their sides is proportional. Write the proportion that relates the vertical leg to the horizontal leg of the triangles. Vertical legs Horizontal Legs = = = Rise Run Note that this proportion shows the slope of line l is the same as the slope of line m. Therefore, parallel lines have the same slope. 4
5 5. Write 2 equations that are parallel to the line y = 2 x Determine which of the following lines are parallel to 2x 3y = 21. Explain why. a. y = 2 3 x + 2 b. 6x + 9y = 12 c. 1 3 x + y = 6. d. 2x + 3y = 7. e. 3x 4.5y = Line m is parallel to the line y = 1 x + 2 and contains the point ( 6,1). What is the equation of 2 Line m in slope-intercept form? 8. What is the equation of the line that passes through (5, 2) and is parallel with a line that passes through (0, 5) and ( 4, 8)? 5
6 Section 2: Perpendicular Lines 1. On a coordinate grid, graph the following pairs of lines a. y = 3 x + 5 and y = 4 x b. y = 3x 1 and y = 1 x 1. 3 c. y = 7x + 2 and y = 1 x 3. 7 d. y = x and y = x 8 Do these lines intersect?. If so, describe the angles formed at their intersection.. Use a protractor if necessary. If not, describe the lines. 6
7 2. Create two equations that have the same type relationship as the lines in Question 1. Draw the lines on a grid to show this relationship. What characteristics do these lines possess?. 3. Will all lines which have these characteristics have the same relationship? If so, prove it. If not, give a counterexample. (Hint: think counterexample) 7
8 4. Use the relationship between slopes of perpendicular lines to answer the following questions. a. Line m has the equation, y = 5 x + 1. What is the slope of a line perpendicular to Line m? 4 b. Write the equation of the line perpendicular to y = 2x + 5 whose y-intercept is 12. c. Write the equation of the line perpendicular to y = 1 x 6 which passes through the point (1, 5 3). d. What is the equation of the line through point (5, 2), and perpendicular to the slope of the line through (0, 5) and (-4, 8) 8
9
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