Lesson 5: Perpendicular Lines
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1 : Perpendicular Lines Learning Target I can generalize the criterion for perpendicularity of two segments that meet at a point I can determine if two segments are perpendicular and write the equation of a line Opening Activity Plot the following points on a coordinate plane: O(0,0), and A( 6,. Rotate the segment OA 90 clockwise round the origin a. Identify the new coordinate of A ( ) and graph segment OA b. What can we say the measure of AOA? c. What can we say about segments OA and OA d. Find the slope of segment OA e. Find the slope of segment OA f. What do you notice about the slopes of segments OA and OA? What is the product of two slopes? In Conclusion.. If two lines are perpendicular then their slopes are opposite reciprocals of each other On the diagram, l 1 l 2 with slopes of m 1 and m 2, then we can say Solve for m 1 m 1 m 2 = 1 m 1 = 1 m 2
2 Example 1. Two lines l 1 and l 2 are perpendicular if the slope of l 1 is 3 what is the slope of l 2? Example 2. What is the slope of a line perpendicular to the line whose equation is y = 3x + 4? Example 3. What is the slope of a line perpendicular to the line whose equation is 5x + 3y = 8? Example 4. What is the equation of the line that passes through the point ( 9,6) and is perpendicular to the line y = 3x 5? Example 5. Find an equation of the line passing through the point (6,5) and perpendicular to the line whose equation is 2y + 3x = 6. Example 6. Points O(0, 0), P(3,, and Q(2, are plotted on the coordinate plane. Determine whether OP and OQ are perpendicular. Support your findings. Example 7. Gerry thinks that the points (4, and ( 1, form a line perpendicular to a line with slope 4. Do you agree? Why or why not?
3 Example 8. Carlos thinks that the segment having endpoints A(0, 0) and B(3, is perpendicular to the segment with endpoints A(0, 0) and C(2,. Do you agree? Why or why not? Definition: A line segment with one endpoint on a line and perpendicular to the line is called a What happens with horizontal and vertical lines, can they be perpendicular? Write the equation of the vertical line Write the equation of the horizontal line Are these two lines perpendicular? l 1 Slope of horizontal line Slope of vertical line If l 1 l 2 then m 1 m 2 = 1 l 2 Conclusion: Vertical and horizontal lines are perpendicular to one another even though the slopes don t fit the perpendicularity Criteria (Perpendicularity criteria states that Slopes are opposite reciprocals when lines are perpendicular)
4 : Perpendicular Lines Exit Ticket ( Fluency ) 1. What is the slope of a line perpendicular to the line whose equation is? 2. Which line is perpendicular to? 3. What is an equation of the line that is perpendicular to the line whose equation is and that passes through the point? 4. Which equation represents the line that is perpendicular to and passes through the point 5. Given A(5, 7) and B(8, : a. Find an equation for the line through A and perpendicular to AB. b. Find an equation for the line through B and perpendicular to AB.
5 : Perpendicular Lines Problem Set 1. Given points S(2,, T(7, 6), U( 3,, and V( 1, 9) Are the lines ST and UV perpendicular? S(2, T(7, 6) U( 3, V( 1, 9) 2. Given U( 4, and V(7, : (use the grid to plot the points) a. Write an equation for the line through U and perpendicular to UV. b. Write an equation for the line through V and perpendicular to UV. 3. Given points C( 4, and D(3, : a. Write the equation of the line through C and perpendicular to CD. b. Write the equation of the line through D and perpendicular to CD.
6 4. Given points N(7, 6) and M(7, : a. Write the equation of the line through M and perpendicular to MN. b. Write the equation of the line through N and perpendicular to MN. 5. Given points A(2,, B(10, 16), C( 3,, and D(4,, are segments AB and CD perpendicular? Are the lines containing the segments perpendicular? 6. Which equation represents a line perpendicular to the line whose equation is? 7. The equation of a line is. What is the slope of the line perpendicular to the given line? 8. What is the slope of a line perpendicular to the line whose equation is? 3
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