Lesson 8: Parallel and Perpendicular Lines

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M4 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.”

Lesson Notes This lesson brings together several of the ideas from the previous lessons. In places where ideas from certain lessons are employed, these are identified with the lesson number underlined (e.g., Lesson 6).

Classwork Opening (5 minutes) Students begin the lesson with the following activity using geometry software to reinforce the theorem studied in

Lesson 6, which states that given points 퐴(푎1,푎2), 퐵(푏1, 푏2), 퐶(푐1, 푐2), and 퐷(푑1, 푑2), ̅퐴퐵̅̅̅ ⊥ ̅퐶퐷̅̅̅ if and only if (푏1 − 푎1)(푑1 − 푐1) + (푏2 − 푎2)(푑2 − 푐2) = 0. . Construct two perpendicular segments, and measure the abscissa (the 푥-coordinate) and ordinate (the 푦-coordinate) of each of the endpoints of the segments. (The teacher may extend this activity by asking students to determine whether the points used must be the endpoints. This is easily investigated using the dynamic geometry software by creating free moving points on the segment and watching the sum of the products of the differences as the points slide along the segments.)

. Calculate (푏1 − 푎1)(푑1 − 푐1) + (푏2 − 푎2)(푑2 − 푐2). . Note the value of the sum, and observe what happens to the sum as students manipulate the endpoints of the perpendicular segments.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M4 GEOMETRY

Example 1 (10 minutes)

Let 푙1and 푙2 be two non-vertical lines in the Cartesian plane. 푙1and 푙2 are perpendicular if and only if their slopes are negative reciprocals of each other (Lesson 5). Explain to students that negative reciprocals also means the product of the slopes is −1.

PROOF:

Suppose 푙1 is given by the equation 푦 = 푚1푥 + 푏1, and 푙2 is given by the equation 푦 = 푚2푥 + 푏2. Start by assuming 푚1, 푚2, 푏1, and 푏2 are all not zero. Students revisit this proof for cases where one or more of these values is zero in the practice problems.

Let’s find two useful points on 푙1: 퐴(0, 푏1) and 퐵(1, 푚1 + 푏1).

. Why are these points on 푙1?  퐴 is the 푦-intercept, and 퐵 is the 푦-intercept plus the slope.

. Similarly, find two useful points on 푙2. Name them 퐶 and 퐷.

 퐶(0, 푏2) and 퐷(1, 푚2 + 푏2). . Explain how you found them and why they are

different from the points on 푙1.

 We used the 푦-intercept of 푙2 to find 퐶 and then added the slope to the 푦-intercept to find point 퐷. They are different points because the lines are different, and the lines do not intersect at those points. . By the theorem from the Opening Exercise (and MP.7 Lesson 5), write the equation that must be satisfied if 푙1 and 푙2 are perpendicular. Take a minute to write your answer, and then explain your answer to your neighbor.

 If 푙1 is perpendicular to 푙2, then we know from the theorem we studied in Lesson 5 and used in our Opening Exercise that this means

(1 − 0)(1 − 0) + (푚1 + 푏1 − 푏1)(푚2 + 푏2 − 푏2) = 0

⟺ 1 + 푚1 ∙ 푚2 = 0

⟺ 푚1푚2 = −1. Summarize this proof and its result to a neighbor (Lesson 5).

If either 푚1 or 푚2 are 0, then 푚1푚2 = −1 is false, meaning 푙1 ⊥ 푙2 is false, that is they cannot be perpendicular. Students study this case later.

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Discussion (Optional) . Working in pairs or groups of three, ask students to use this new theorem to construct an explanation of why

(푏1 − 푎1)(푑1 − 푐1) + (푏2 − 푎2)(푑2 − 푐2) = 0 when ̅퐴퐵̅̅̅ ⊥ ̅퐶퐷̅̅̅ (Lesson 6). As students construct their arguments, circulate around the room listening to the progress that is being made with an eye to strategically selecting pairs or groups to share their explanations with the whole group in a manner that builds from basic arguments that may not be fully formed to more sophisticated and complete explanations. The explanations should include the following understandings:

푏2− 푎2 푑2− 푐2  푚퐴퐵 = and 푚퐶퐷 = 푏1− 푎1 푑1− 푐1

 Because we constructed the segments to be perpendicular, we know that 푚퐴퐵푚퐶퐷 = −1 (Lesson 7). 푏2− 푎2 푑2− 푐2  푚퐴퐵푚퐶퐷 = −1 ⟺ ∙ = −1 ⟺ (푏2 − 푎2)(푑2 − 푐2) = −(푏1 − 푎1)(푑1 − 푐1) 푏1− 푎1 푑1− 푐1

⟺ (푏1 − 푎1)(푑1 − 푐1) + (푏2 − 푎2)(푑2 − 푐2) = 0 (Lesson 6)

Exercise 1 (5 minutes)

Scaffolding: Exercise 1 . If students are struggling, 1. change this example to a. Write an equation of the line that passes through the origin and intersects the line specific lines. Give the ퟐ풙 + ퟓ풚 = ퟕ to form a right angle. coordinates of the two ퟓ parallel lines and the 풚 = 풙 ퟐ perpendicular line. Calculate the slopes of ퟑ each and compare. ퟐ풙 + ퟑ풚 = ퟔ 풚 = 풙 + ퟒ b. Determine whether the lines given by the equations and ퟐ are perpendicular. Support your answer. . The teacher may also have ퟐ ퟑ students construct two − The slope of the first line is ퟑ, and the slope of the second line is ퟐ. The product of lines that are parallel to a these two slopes is −ퟏ; therefore, the two lines are perpendicular. given line using dynamic geometry software and c. Two lines having the same 풚-intercept are perpendicular. If the equation of one of then measure and ퟒ 풚 = − 풙 + ퟔ compare their slopes. these lines is ퟓ , what is the equation of the second line? ퟓ 풚 = 풙 + ퟔ ퟒ

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Example 2 (12 minutes)

In this example, students study the relationship between a pair of parallel lines and the lines that they are perpendicular to. They use the angle congruence axioms developed in Geometry Module 4 for parallelism. Students are investigating two questions in this example. This example can be done using dynamic geometry software, or students can just sketch the lines and note the angle pair relationships.

Example 2

a. What is the relationship between two coplanar lines that are perpendicular to the same line?

Have students draw a line and label it 푘. Students then construct line 푙1 perpendicular to line 푘. Finally, students construct line 푙2 not coincident with line 푙1, also perpendicular to line 푘.

. What can we say about the relationship between lines 푙1 and 푙2?  These lines are parallel because the corresponding angles that are created by the transversal 푘 are congruent.

. If the slope of line 푘 is 푚, what is the slope of 푙1? Support your answer. (Lesson 5) 1  The slope of 푙 is − because the slopes of perpendicular lines 1 푚 are negative reciprocals of each other.

. If the slope of line 푘 is 푚, what is the slope of 푙2? Support your answer. (Lesson 5) 1  The slope of 푙 is − because the slopes of perpendicular lines 2 푚 are negative reciprocals of each other.

. Using your answers to the last two questions, what can we say about the slopes of lines 푙1 and 푙2?

 푙1 and 푙2 have equal slopes.

. What can be said about 푙1 and 푙2 if they have equal slopes?

 푙1 and 푙2 are parallel. . What is the relationship between two coplanar lines that are perpendicular to the same line? MP.3  If two lines are perpendicular to the same line, then the two lines are parallel, and if two lines are parallel, then their slopes are equal. . Restate this to your partner in your own words, and explain it by drawing a picture.

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b. Given two lines, 풍ퟏ and 풍ퟐ, with equal slopes and a line 풌 that is perpendicular to one of these two parallel lines, 풍ퟏ: i. What is the relationship between line 풌 and the other line,

풍ퟐ?

ii. What is the relationship between 풍ퟏ and 풍ퟐ?

Ask students to do the following: . Construct two lines that have the same slope using construction tools or dynamic geometry software and label

the lines 푙1 and 푙2. . Construct a line that is perpendicular to one of these lines and label this line 푘.

. If 푙2 has a slope of 푚, what is the slope of line 푘? Support your answer. (Lesson 5) 1  Line 푘 must have a slope of − because line 푘 is perpendicular to line 푙 , and the slopes of 푚 2 perpendicular lines are negative reciprocals of each other.

. 푙1 also has a slope of 푚, so what is its relationship to line 푘? Support your answer. (Lesson 5)

 Line 푙1 must be perpendicular to line 푘 because their slopes are negative reciprocals of each other. . Using your answers to the last two questions, what can we say about two lines that have equal slopes?  They must be parallel since they are both perpendicular to the same line. Have students combine the results of the two activities to form the following bi-conditional statement: Two non-vertical lines are parallel if and only if their slopes are equal. . Summarize all you know about the relationships between slope, parallel lines, and perpendicular lines. Share your ideas with your neighbor.

Exercises 2–7 (8 minutes)

Exercises 2–7

2. Given a point (−ퟑ, ퟔ) and a line 풚 = ퟐ풙 − ퟖ: a. What is the slope of the line?

ퟐ

b. What is the slope of any line parallel to the given line?

ퟐ

c. Write an equation of a line through the point and parallel to the line.

ퟐ풙 − 풚 = −ퟏퟐ

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d. What is the slope of any line perpendicular to the given line? Explain. ퟏ − The slope is ퟐ; perpendicular lines have slopes that are negative reciprocals of each other.

ퟏ (ퟎ, −ퟕ) 풚 = 풙 + ퟓ 3. Find an equation of a line through and parallel to the line ퟐ . a. What is the slope of any line parallel to the given line? Explain your answer. ퟏ

The slope is ퟐ; lines are parallel if and only if they have equal slopes.

b. Write an equation of a line through the point and parallel to the line.

풙 − ퟐ풚 = ퟏퟒ

ퟏ 풚 = 풙 + ퟓ 풙 − ퟐ풚 = ퟏퟒ c. If a line is perpendicular to ퟐ , will it be perpendicular to ? Explain. ퟏ ퟏ 풚 = 풙 + ퟓ 풚 = 풙 − ퟕ If a line is perpendicular to ퟐ , it will also be perpendicular to ퟐ because a line perpendicular to one line is perpendicular to all lines parallel to that line.

ퟏ ( ퟑ, ) 4. Find an equation of a line through √ ퟐ parallel to the line: a. 풙 = −ퟗ

풙 = √ퟑ

b. 풚 = −√ퟕ ퟏ 풚 = ퟐ

c. What can you conclude about your answer in parts (a) and (b)? ퟏ 풙 = ퟑ 풚 = They are perpendicular to each other. √ is a vertical line, and ퟐ is a horizontal line.

5. Find an equation of a line through (−√ퟐ, 흅) parallel to the line 풙 − ퟕ풚 = √ퟓ.

풙 − ퟕ풚 = −√ퟐ − ퟕ흅

6. Recall that our search robot is moving along the line 풚 = ퟑ풙 − ퟔퟎퟎ and wishes to make a right turn at the point (ퟒퟎퟎ, ퟔퟎퟎ). Find an equation for the perpendicular line on which the robot is to move. Verify that your line intersects the 풙-axis at (ퟐퟐퟎퟎ, ퟎ).

풙 + ퟑ풚 = ퟐퟐퟎퟎ; when 풚 = ퟎ, 풙 = ퟐퟐퟎퟎ

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7. A robot, always moving at a constant speed of ퟐ units per second, starts at position (ퟐퟎ, ퟓퟎ) on the coordinate plane and heads in a southeast direction along the line ퟑ풙 + ퟒ풚 = ퟐퟔퟎ. After ퟏퟓ seconds, it turns clockwise ퟗퟎ° 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?

The coordinates are (ퟒퟒ, ퟑퟐ). We know the robot moves down ퟑ units and right ퟒ units, and the distance moved in ퟏퟓ seconds is ퟑퟎ units. This gives us a right triangle with hypotenuse ퟓ, so we need to move this way ퟔ times. From (ퟐퟎ, ퟓퟎ), move a total of down ퟏퟖ units and right ퟐퟒ units.

b. Find an equation for the second line on which the robot traveled.

ퟒ풙 − ퟑ풚 = ퟖퟎ

c. If, after turning, the robot travels for ퟐퟎ seconds along this line and then stops, how far will it be from its starting position?

It will be at the point (ퟐퟎ, ퟎ), ퟓퟎ units 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?

풙 = ퟐퟎ; ퟐퟓ seconds

Closing (2 minutes) Ask students to respond to these questions in writing, with a partner, or as a class. . Samantha claims the slopes of perpendicular lines are opposite reciprocals, while Jose says the product of the slopes of perpendicular lines is equal to −1. Discuss these two claims (Lesson 5).  Both are correct as long as the lines are not horizontal and vertical. If two lines have negative 1 1 reciprocal slopes, 푚 and − , the product 푚 ∙ − = −1. 푚 푚  If one of the lines is horizontal, then the other will be vertical, and the slopes will be zero and undefined, respectively. Therefore, these two claims will not hold. . How did we use the relationship between perpendicular lines to demonstrate the theorem two non-vertical lines are parallel if and only if their slopes are equal, and why did we restrict this statement to non-vertical lines (Lesson 8)?  We restricted our discussion to non-vertical lines because the slopes of vertical lines are undefined.  We used the fact that the slopes of perpendicular lines are negative reciprocals of each other to show that if two lines are perpendicular to the same line, then they must not only be parallel to each other, but also have equal slopes.

Exit Ticket (3 minutes) Teachers can administer the Exit Ticket in several ways, such as the following: assign the entire task, assign some students Problem 1 and others Problem 2, and allow student choice.

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Name Date

Lesson 8: Parallel and Perpendicular Lines

Exit Ticket

1. Are the pairs of lines parallel, perpendicular, or neither? Explain. a. 3푥 + 2푦 = 74 and 9푥 − 6푦 = 15

b. 4푥 − 9푦 = 8 and 18푥 + 8푦 = 7

2. Write the equation of the line passing through (−3, 4) and perpendicular to −2푥 + 7푦 = −3.

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Exit Ticket Sample Solutions

1. Are the pairs of lines parallel, perpendicular, or neither? Explain. a. ퟑ풙 + ퟐ풚 = ퟕퟒ and ퟗ풙 − ퟔ풚 = ퟏퟓ ퟑ ퟑ − The lines are neither, and the slopes are ퟐ and ퟐ.

b. ퟒ풙 − ퟗ풚 = ퟖ and ퟏퟖ풙 + ퟖ풚 = ퟕ ퟒ ퟗ − The lines are perpendicular, and the slopes are ퟗ and ퟒ.

2. Write the equation of the line passing through (−ퟑ, ퟒ) and normal to −ퟐ풙 + ퟕ풚 = −ퟑ. ퟕ풙 + ퟐ풚 = −ퟏퟑ

Problem Set Sample Solutions

1. Write the equation of the line through (−ퟓ, ퟑ) and: a. Parallel to 풙 = −ퟏ.

풙 = −ퟓ

b. Perpendicular to 풙 = −ퟏ.

풚 = ퟑ

ퟑ 풚 = 풙 + ퟐ c. Parallel to ퟓ . ퟑ풙 − ퟓ풚 = −ퟑퟎ

ퟑ 풚 = 풙 + ퟐ d. Perpendicular to ퟓ . ퟓ풙 + ퟑ풚 = −ퟏퟔ

ퟓ ( ퟑ, ) 2. Write the equation of the line through √ ퟒ and: a. Parallel to 풚 = ퟕ. ퟓ 풚 = ퟒ

b. Perpendicular to 풚 = ퟕ.

풙 = √ퟑ

ퟏ ퟑ c. Parallel to 풙 − 풚 = ퟏퟎ. ퟐ ퟒ ퟖ풙 − ퟏퟐ풚 = −ퟏퟓ + ퟖ√ퟑ

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ퟏ ퟑ d. Perpendicular to 풙 − 풚 = ퟏퟎ. ퟐ ퟒ ퟔ풙 + ퟒ풚 = ퟓ + ퟔ√ퟑ

3. A vacuum robot is in a room and charging at position (ퟎ, ퟓ). Once charged, it begins moving on a northeast path at ퟏ a constant speed of foot per second along the line ퟒ풙 − ퟑ풚 = −ퟏퟓ. After ퟔퟎ seconds, it turns right ퟗퟎ° and ퟐ travels in the new direction. a. What are the coordinates of the point at which the robot made the turn?

(ퟏퟖ, ퟐퟗ)

b. Find an equation for the second line on which the robot traveled.

ퟑ풙 + ퟒ풚 = ퟏퟕퟎ

c. If after turning, the robot travels ퟖퟎ seconds along this line, what is the distance between the starting position and the robot’s current position?

ퟓퟎ feet

d. What is the equation of the line the robot needs to travel along in order to return and recharge? How long will it take the robot to get there?

풚 = ퟓ; ퟏퟎퟎ seconds

4. Given the statement 푨푩⃡ is parallel to 푫푬⃡ , construct an argument for or against this statement using the two triangles shown.

The statement 푨푩⃡ is parallel to 푫푬⃡ is true. ퟐ The slope of 푨푩⃡ is equal to , the ratio of the lengths of the ퟔ legs of the right triangle 푨푩푪. ퟏ The slope of 푫푬⃡ is equal to , the ratio of the lengths of the ퟑ legs of the right triangle 푫푬푭. ퟐ ퟏ = ↔ 풎 = 풎 , and therefore, 푨푩⃡ is parallel to 푫푭⃡ . ퟔ ퟑ 푨푩 푫푬

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5. Recall the proof we did in Example 1: Let 풍ퟏ and 풍ퟐ be two non-vertical lines in the Cartesian plane. 풍ퟏand 풍ퟐ are perpendicular if and only if their slopes are negative reciprocals of each other. In class, we looked at the case where both 풚-intercepts were not zero. In Lesson 5, we looked at the case where both 풚-intercepts were equal to zero, when the vertex of the right angle was at the origin. Reconstruct the proof for the case where one line has a 풚-intercept of zero, and the other line has a nonzero 풚-intercept.

Suppose 풍ퟏ passes through the origin; then, it will be given by the equation 풚 = 풎ퟏ풙, and 풍ퟐ Scaffolding: is given by the equation 풚 = 풎ퟐ풙 + 풃ퟐ. Here, we are assuming 풎ퟏ, 풎ퟐ, and 풃ퟐ are not zero. Provide students who were absent or who do not take We can still use the same two points on 풍ퟐ as we used in Example 1, (ퟎ, 풃ퟐ) and (ퟏ, 풎ퟐ + 풃ퟐ). We will have to find an additional point to use on 풍ퟏ, as the 풙- and 풚- notes well with class notes intercepts are the same point because this line passes through the origin, (ퟎ, ퟎ). Let’s let showing Example 1 with steps our second point be (ퟏ, 풎ퟏ). clearly explained.

If 풍ퟏ ⊥ 풍ퟐ then:

(ퟏ − ퟎ)(ퟏ − ퟎ) + (풎ퟐ − ퟎ)(풃ퟐ − ퟎ) = ퟎ

ퟏ + 풎ퟏ풎ퟐ = ퟎ

ퟏ = −풎ퟏ풎ퟐ ퟏ − = 풎ퟐ 풎ퟏ

6. Challenge: Reconstruct the proof we did in Example 1 if one line has a slope of zero.

If 풎ퟏ = ퟎ, then 풍ퟏ is given by the equation 풚 = 풃ퟏ, and 풍ퟐ is given by the equation 풚 = 풎ퟐ풙 + 풃ퟐ. Here we are assuming 풎ퟐ, 풃ퟏ, and 풃ퟐ are not zero.

Choosing the points (ퟎ, 풃ퟏ) and (ퟏ, 풃ퟏ) on 풍ퟏ and (ퟎ, 풃ퟐ) and (ퟏ, 풎ퟐ + 풃ퟐ) on 풍ퟐ,

If 풍ퟏ ⊥ 풍ퟐ, then:

(ퟏ − ퟎ)(ퟏ − ퟎ) + (풃ퟏ − 풃ퟏ)(풎ퟐ + 풃ퟐ − 풃ퟐ) = ퟎ ퟏ + ퟎ = ퟎ ퟏ = ퟎ

This is a false statement. If one of the perpendicular lines is horizontal, their slopes cannot be negative reciprocals.

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