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Name ______________________________ Equations of Parallel and Perpendicular Lines Module 4 Learning Target: I can apply slope criteria for parallel and perpendicular lines and use them to solve geometric problems. Opening Exercises 1. What is the slope of the line passing through the points A and B, as shown on the graph below? 2. Write the equation of the line graphed below. Equations of Parallel and Perpendicular Lines Essential Understanding: Parallel lines have equal slopes. 1. Determine the slope of each line to show that the lines are parallel. 2. Which equation represents a line parallel to the graph of 2𝑥 − 4𝑦 = 16? 1 (1) 𝑦 = 2 𝑥 − 5 1 (2) 𝑦 = − 2 𝑥 + 4 (3) 𝑦 = −2𝑥 + 6 (4) 𝑦 = 2𝑥 + 8 3. Which equation represents a line parallel to the x-axis? (1) (2) (3) (4) 𝑥=5 𝑦 = 10 1 𝑥 = 3𝑦 𝑦 = 5𝑥 + 17 4. Write an equation of the line that passes through the point (4,2) that is parallel to the line whose equation is 3𝑥 + 6𝑦 = 12. Essential Understanding: The slopes of perpendicular lines are negative reciprocals. 5. What is the slope of a line perpendicular to the line 𝑦 = 5𝑥 − 4? 6. Write an equation of the line that passes through the point (4,2) that is perpendicular to the line whose equation is 3𝑥 + 6𝑦 = 12. 7. Find the distance between the point (0,0) and the line 𝑦 = −𝑥 + 4. Essential Understanding: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. A. Graph the equation on the line that passes through the points 𝐴(−4,8) and 𝐵(4,6). B. What is the equation of the line passing through the points A and B, as shown on the graph above? C. The line that passes through the points A and B is dilated by a scale factor of 1 2 and centered at the origin. What is the equation that represents the image of the line after the dilation? Name ______________________________ Equations of Parallel and Perpendicular Lines Module 4 Problem Set 1. Which equation represents a line parallel to the y-axis? (1) (2) (3) (4) 𝑥=𝑦 𝑥=4 𝑦=4 𝑦 =𝑥+4 2. Which equation represents a line that is parallel to the line 𝑦 = 3 − 2𝑥? (1) (2) (3) (4) 4𝑥 + 2𝑦 = 5 2𝑥 + 4𝑦 = 1 𝑦 = 3 − 4𝑥 𝑦 = 4𝑥 − 2 3 3. The line y = 2x – 4 is dilated by a scale factor of 2 and centered at the origin. Which equation represents the image of the line after the dilation? (1) y = 2x – 4 (2) y = 2x – 6 (3) y = 3x – 4 (4) y = 3x – 6 4. The equation of line h is 2𝑥 + 𝑦 = 1. Line m is the image of line h after a dilation of scale factor 4 with respect to the origin. What is the equation of the line m? (1) y = –2x + 1 (2) y = –2x + 4 (3) y = 2x + 4 (4) y = 2x + 1 5. If the graphs of the equations 𝑦 − 2𝑥 and 𝑦 − 𝑘𝑥 = 7 are parallel, what is the value of k? 6. Write the equation of a line that passes through the point (−5,3) and is perpendicular 3 to 𝑦 = 5 𝑥 + 2. 5 7. Write the equation of a line that passes through the point (√3, 4) and is parallel 1 3 to 2 𝑥 − 4 𝑦 = 10. 8. Are the lines 4𝑥 − 9𝑦 = 8 and 18𝑥 + 8𝑦 = 7 parallel, perpendicular, or neither? Explain. 9. Are the lines 3𝑥 + 2𝑦 = 74 and 9𝑥 − 6𝑦 = 15 parallel, perpendicular, or neither? Explain. 10. Line 𝐴 contains points (𝑝−4,2) and (−2,9). Line 𝐵 contains points (𝑝,−1) and (−1,1). a. Find the value of 𝑝 if the lines are parallel. b. Find the value(s) of 𝑝 if the lines are perpendicular. 11. Find the distance between the point (0,0) and the line 𝑦 = 𝑥 + 10. Name ______________________________ Equations of Parallel and Perpendicular Lines Write the equation of the line that contains the point (−2,7) and is a. Parallel to 𝑥 = 3. b. Perpendicular to 𝑥 = −3. c. Parallel to 𝑦 = 6𝑥 − 13. d. Perpendicular to 𝑦 = 6𝑥 − 13. Module 4 Exit Ticket