Parallel and Perpendicular Lines

New York State Common Core Math Geometry, Module 4, Lesson 8

Worksheets for Geometry

Student Outcomes

• Students recognize parallel and perpendicular lines from slope.
• Students create equations for lines satisfying criteria of the kind: “Contains a given point and is parallel/perpendicular to a given line.”

Parallel and Perpendicular Lines

Classwork

Opening Exercise

1. a. Write an equation of the line that passes through the origin that intersects the line 2𝑥 + 5𝑦 = 7 to form a right angle.
   b. Determine whether the lines given by the equations 2𝑥 +3𝑦 = 6 and 𝑦 = 3/2 𝑥 + 4 are perpendicular. Support your answer.
   c. Two lines having the same 𝑦-intercept are perpendicular. If the equation of one of these lines is 𝑦 = −4/5 𝑥 + 6, what is the equation of the second line?

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Example 1

a. What is the relationship between two coplanar lines that are perpendicular to the same line?
b. Given two lines, 𝑙1 and 𝑙2, with equal slopes and a line 𝑘 that is perpendicular to one of these two parallel lines, 𝑙1:
i. What is the relationship between line 𝑘 and the other line, 𝑙2?
ii. What is the relationship between 𝑙1 and 𝑙2?

Exercises 2–7

2. Given a point (−3,6) and a line 𝑦 = 2𝑥 − 8:
   a. What is the slope of the line?
   b. What is the slope of any line parallel to the given line?
   c. Write an equation of a line through the point and parallel to the line.
   d. What is the slope of any line perpendicular to the given line? Explain.
3. Find an equation of a line through (0,−7) and parallel to the line 𝑦 = 1/2𝑥 + 5.
   a. What is the slope of any line parallel to the given line? Explain your answer.
   b. Write an equation of a line through the point and parallel to the line.
   c. If a line is perpendicular to 𝑦 = 1/2 𝑥 + 5, will it be perpendicular to 𝑥 −2𝑦 = 14? Explain.
4. Find an equation of a line through (√3,1/2) parallel to the line:
   a. 𝑥 = −9
   b. 𝑦 = −√7
   c. What can you conclude about your answer in parts (a) and (b)?
5. Find an equation of a line through (−√2, 𝜋) parallel to the line 𝑥 − 7𝑦 = √5.
   
6. Recall that our search robot is moving along the line 𝑦 = 3𝑥 − 600 and wishes to make a right turn at the point (400,600). Find an equation for the perpendicular line on which the robot is to move. Verify that your line intersects the 𝑥-axis at (2200,0).
7. A robot, always moving at a constant speed of 2 units per second, starts at position (20,50) on the coordinate plane and heads in a southeast direction along the line 3𝑥 + 4𝑦 = 260. After 15 seconds, it turns clockwise 90° and travels in a straight line in this new direction.
   a. What are the coordinates of the point at which the robot made the turn? What might be a relatively straightforward way of determining this point?
   b. Find an equation for the second line on which the robot traveled.
   c. If, after turning, the robot travels for 20 seconds along this line and then stops, how far will it be from its starting position?
   d. What is the equation of the line the robot needs to travel along in order to now return to its starting position? How long will it take for the robot to get there?

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