GEOMETRY MODULE 4 LESSON 2 PARALLEL AND PERPENDICULAR LINES OPENING EXERCISE Plot the points � 0,0 , � 3, −1 , � (2,3) on the coordinate plane. Determine whether �� and �� are perpendicular. Explain. If we can show that the triangle formed be the segments is a right triangle, we can say that �� and �� are perpendicular. Using the distance formula, �� = 10, �� = 13, �� = 17. ! ! 10 + 13 ≠ 17 ! So, �� and �� are not perpendicular. CRITERIA FOR PARALLEL AND PERPENDICULAR LINES It’s all about the slope! • The slopes of parallel lines are equal. �! = �! ! • The slopes of perpendicular lines are negative reciprocal of each other. �! = − !! What type of angle do perpendicular lines form? The perpendicular lines form a right (90°) angle. TESTING FOR PERPENDICULARITY • Find and compare the slopes of each segment/line created by the given points. • Determine if the condition �!�! + �!�! = 0 is satisfied. Note: Segments may need to be translated so that one endpoint is on the origin (0,0). MOD4 L2 1 Apply each test to the points from the opening exercise. • Find and compare slopes. �! − �! 3 − 0 3 �����!" = = = �! − �! 2 − 0 2 �! − �! −1 − 0 −1 �����!" = = = �! − �! 3 − 0 3 • Test �!�! + �!�! = 0 2 ∙ 3 + 3 ∙ −1 = 6 + −3 ≠ 0 PRACTICE • Are the pairs of lines parallel, perpendicular, or neither? 3� + 2� = 74 ��� 9� − 6� = 15 Convert each equation to slope-intercept form and compare slopes. 3 3� + 2� = 74 → � = − � + 37 2 3 5 9� − 6� = 15 → � = � − 2 2 NEITHER 4� − 9� = 8 ��� 18� + 8� = 7 4 8 4� − 9� = 8 → � = � − 9 9 9 7 18� + 8� = 7 → � = − � + 4 8 PERPENDICULAR ON YOUR OWN −4� + 5� = −35 ��� − 8� + 10� = 200 4 −4� + 5� = −35 → � = � − 7 5 4 −8� + 10� = 200 → � = � + 20 5 PARALLEL MOD4 L2 2 PRACTICE • Write the equation of the line passing through (−3, 4) and is perpendicular to−2� + 7� = −3. • Write the equation of the line passing through (−3, 4) and is parallel to−2� + 7� = −3. Convert the given equation to slope-intercept form. 2 3 � = � − 7 7 The slope of this line is !, thus the perpendicular slope is − ! and the parallel slope is !. ! ! ! � − �! = �(� − �!) � − �! = �(� − �!) 7 2 � − 4 = − (� − (−3)) � − 4 = (� − (−3)) 2 7 7 21 2 6 � − 4 = − � − � − 4 = � + 2 2 7 7 7 21 8 2 6 28 � = − � − + � = � + + 2 2 2 7 7 7 7 13 2 34 � = − � − � = � + 2 2 7 7 VERTICAL AND HORIZONTAL LINES The equation � = 2 refers to a position on the x-axis. Note that the y-variable is not present in the equation. Thus, y can be any value. • The equation � = 2 represents a vertical line. • Its slope is undefined. • Find an equation of a line that goes through the point (−4, 7) and is parallel to � = 2. � = −4 MOD4 L2 3 The equation � = 3 refers to a position on the y-axis. Note that the x-variable is not present in the equation. Thus, x can be any value. • The equation � = 3 represents a horizontal line. • Its slope is zero. • Find an equation of a line that goes through the point (−4, 7) and is parallel to � = 3. � = 7 • How would the x-axis be represented in an equation? � = 0 • How would the y-axis be represented in an equation? � = 0 • Find an equation of a line that goes through (−4, 7) and is perpendicular to � = 2. � = 7 MOD4 L2 4 .