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Lesson 7 NYS COMMON CORE MATHEMATICS CURRICULUM M4 GEOMETRY Lesson 7: Equations for Lines Using Normal Segments Student Outcomes ο§ Students recognize a segment perpendicular to a line, with one of its endpoints on the line as a normal segment. ο§ Students recognize that when a line and a normal segment intersect at the origin, the segment from (0,0) to (π1 , π2 ) is the normal segment, with a slope of of β π1 . π2 π2 π1 , and the equation of the line is π1 π₯ + π2 π¦ = π with a slope Lesson Notes This lesson focuses on MP.4 because students work extensively to model robot behavior using coordinates. Classwork Opening Exercise (5 minutes) This exercise can be modeled by the teacher with the whole class, given to groups to present solutions to the class, or used as a supplement to the lesson. Opening Exercise The equations given are in standard form. Put each equation in slope-intercept form. State the slope and the π-intercept. 1. ππ + ππ = ππ 2. ππ + ππ = ππ π = βππ + π π π= β π+π π slope = βπ slope = β π-intercept = π π-intercept = π π π 3. ππ β ππ = βπ π= π π π+ π π slope = π π π-intercept = Scaffolding: π π Provide visuals to reinforce standard and slope-intercept forms: ο§ Standard form is π΄π₯ + π΅π¦ = πΆ where π΄, π΅, and πΆ are integers. ο§ Slope-intercept form is π¦ = ππ₯ + π where π is the slope, and π is the π¦intercept. Lesson 7: Equations for Lines Using Normal Segments This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M4-TE-1.3.0-09.2015 72 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 7 NYS COMMON CORE MATHEMATICS CURRICULUM M4 GEOMETRY Discussion (20 minutes) Letβs revisit the robot from Lesson 4. Recall that it is moving along the line π¦ = 3π₯ β 600. At the point (400, 600), it detected the loudest βping,β and the programmers had the robot change direction at that point and move along a linear path that was perpendicular to its original path. Scaffolding: Plotting points helps students visualize the problems and understand the translations required. At the end of Lesson 6, students theorized that the beacon might lie on the π₯-axis. If it did, it would be located at the point (2200, 0). ο§ How did we determine the location of the beacon? We translated point πΉ to the origin and substituted the coordinates of the translated points πΈβ²and πΊβ² into the formula for the condition of perpendicularity π1 π1 + π2 π2 = 0, using π β 400 as the π₯-coordinate of πΊ β² . We solved the equation to get π = 2200. Letβs push this idea a little further. Suppose π(π₯, π¦) is any point on the line containing Μ Μ Μ Μ πΈπΉ . What can you say Μ Μ Μ Μ and Μ Μ Μ Μ about πΉπ πΉπΊ ? οΊ ο§ οΊ ο§ Using πΉ(400, 600), πΊ(2200, 0), and π(π₯, π¦), translate the points so that πΉ is at the origin. What are the new coordinates? οΊ ο§ They are also perpendicular. If we translate all points in the figure using the translation vector β©β400, β600βͺ that takes πΉ to the origin, then πΉβ² is (0, 0), πΊ becomes πΊβ²(1800, β600), and π becomes πβ²(π₯ β 400, π¦ β 600). If the condition for perpendicularity is ππ + ππ = 0, how could we write an equation involving π₯ and π¦? οΊ The condition that Μ Μ Μ Μ Μ πΉβ²πΊβ² and Μ Μ Μ Μ Μ πΉβ²πβ² are perpendicular becomes: 1800(π₯ β 400) + (β600)(π¦ β 600) = 0 1800π₯ β 720,000 β 600π¦ + 360,000 = 0 1800π₯ β 600π¦ = 360,000. ο§ This is the equation of the line in standard form: π΄π₯ + π΅π¦ = πΆ. Look at the values of π΄ and π΅ and at the work above. Do you see a relationship between π΄ and π΅ and the work above? οΊ The coordinates of πΊβ² (the translation of πΊ) are (1800, β600), which are the same values as π΄ and π΅. Lesson 7: Equations for Lines Using Normal Segments This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M4-TE-1.3.0-09.2015 73 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 7 NYS COMMON CORE MATHEMATICS CURRICULUM M4 GEOMETRY ο§ ο§ Letβs say that in a more specific way. I will state the relationship, and you repeat it and explain it to your partner. οΊ π΄ is the π₯-coordinate (abscissa) of the image of the point of the perpendicular segment that does not lie on the line when the point of perpendicularity is at the origin. οΊ π΅ is the π¦-coordinate (ordinate) of the image of the point of the perpendicular segment that does not lie on the line when the point of perpendicularity is at the origin. Now, letβs put the equation in slope-intercept form. οΊ β600π¦ = β1800π₯ + 360,000 π¦ = 3π₯ β 600, which is the equation of β‘πΈπΉ . ο§ MP.7 What does the equation you wrote represent? οΊ Μ Μ Μ Μ πΉπ will be perpendicular to Μ Μ Μ Μ πΉπΊ as long as point π lies on the line containing Μ Μ Μ Μ πΈπΉ , which is given by the equation π¦ = 3π₯ β 600. In the next part of this lesson, students generalize what they just discovered. This can be done in several ways. 1. Present the question, and explain that they are trying to make this process work for any points with coordinates π΄(π, π), π΅(π, π), and π(π₯, π¦). Allow time for students to think and talk to their neighbors for a few minutes; then, show the diagrams and give them more time to talk. Finally, pull everyone together, and discuss each step as a class. 2. Assign some groups the task with no leading questions, and let them work independently while other groups are getting different levels of help, some even being directly instructed by the teacher. ο§ How can we generalize this finding? ο§ Given point π΄(π, π), which lies on line π; point π΅(π, π) not on line π; and Μ Μ Μ Μ π΄π΅ perpendicular to line π, then any Μ Μ Μ Μ β₯ Μ Μ Μ Μ point π(π₯, π¦) on line π will satisfy the relationship π΄π π΄π΅ . Draw the picture described. π© π (π, π ) (π, π) π· π¨ Lesson 7: (π, π) Equations for Lines Using Normal Segments This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M4-TE-1.3.0-09.2015 74 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 7 NYS COMMON CORE MATHEMATICS CURRICULUM M4 GEOMETRY ο§ Translate the points. Which point should be on the origin? What is the translation used? What are the coordinates of the translated points? οΊ π΄ is the common point and should be translated to the origin. The translation is β©βπ, βπβͺ, or left π π’πππ‘π and down π π’πππ‘π . The translated points are π΄β²(0, 0), π΅β²(π β π, π β π), and πβ²(π₯ β π, π¦ β π). π π©β² (π β π, π β π) π·β² (π β π, π β π) π¨β² ο§ If the segments are perpendicular, write the equation that must hold true. οΊ ο§ ο§ ο§ (π, π) (π β π)(π₯ β π) + (π β π)(π¦ β π) = 0 If π΄ = (π β π) and π΅ = (π β π), write this equation substituting in π΄ and π΅. Which line have we written the equation of? Scaffolding: οΊ We end up with the equation of the line that passes through ο§ Have students leave the equations point π΄ that is perpendicular to Μ Μ Μ Μ π΄π΅ : π΄(π₯ β π) + π΅(π¦ β π) = 0. in standard form. What do π΄ and π΅ represent graphically? ο§ Provide these steps: οΊ π΄ is the abscissa, and π΅ is the ordinate of the image of point π΅. (8 β 5)(π₯ β 5) + (2 + 7)(π¦ + 7) = 0 We call Μ Μ Μ Μ π΄π΅ a normal segment to line π because it has one endpoint on the line and is perpendicular to the line. Simplify the parentheses without variables. οΊ Explain to your neighbor what a normal segment is, and write your own definition. DEFINITION: A line segment with one endpoint on a line and perpendicular to the line is called a normal segment to the line. 3(π₯ β 5) + 9(π¦ + 7) = 0 Separate the variables by putting π₯ on one side and π¦ on the other. 9(π¦ + 7) = β3(π₯ β 5) Distribute the coefficients. 9π¦ + 63 = β3π₯ + 15 Bring the constant on the right to the left. 9π¦ = β3π₯ β 48 Divide by the coefficient of π¦. 3 48 π¦=β π₯β 9 9 Simplify if necessary. 1 16 π¦=β π₯β 3 3 Lesson 7: Equations for Lines Using Normal Segments This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M4-TE-1.3.0-09.2015 75 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M4 GEOMETRY Example (5 minutes) Example Given π¨(π, βπ) and π©(π, π): a. Μ Μ Μ Μ . Find an equation for the line through π¨ and perpendicular to π¨π© π¨β² (π, π), π©β² (π β π, π β (βπ)), and π·β² (π β π, π β (βπ)) (π β π)(π β π) + (π + π)(π + π) = π π(π β π) = βπ(π + π) ππ β ππ = βππ β ππ π ππ π= β πβ π π b. Μ Μ Μ Μ . Find an equation for the line through π© and perpendicular to π¨π© π©β² (π, π), π¨β² (π β π, βπ β π), and π·β²(π β π, π β π) (π β π)(π β π) + (βπ β π)(π β π) = π βπ(π β π) = π(π β π) βππ + ππ = ππ β ππ π ππ π=β π+ π π Exercises (8 minutes) Have students plot points to aid in solving problems. Exercises 1. Given πΌ(βπ, βπ) and π½(π, π): a. Μ Μ Μ Μ . Write an equation for the line through πΌ and perpendicular to πΌπ½ πΌβ² (π, π), π½β² (π β (βπ), π β (βπ)), π·β² (π β (βπ), π β (βπ)) (π + π)(π + π) + (π + π)(π + π) = π πππ + ππ = βππ β π ππ π=β π β ππ π b. Μ Μ Μ Μ . Write an equation for the line through π½ and perpendicular to πΌπ½ π½β² (π, π), πΌβ² (βπ β π, βπ β π), π·β²(π β π, π β π) (βπ β π)(π β π) + (βπ β π)(π β π) = π βπππ + ππ = ππ β π ππ ππ π=β π+ π π Lesson 7: Equations for Lines Using Normal Segments This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M4-TE-1.3.0-09.2015 76 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M4 GEOMETRY 2. Given πΊ(π, βπ) and π»(βπ, ππ): a. Write an equation for the line through πΊ and perpendicular to Μ Μ Μ Μ πΊπ». πΊβ² (π, π), π»β² (βπ β π, ππ β (βπ)), π·β² (π β π, π β (βπ)) (βπ β π)(π β π) + (ππ β (βπ))(π β (βπ)) = π βπππ + ππ = βπππ β ππ ππ πππ π= πβ ππ ππ b. Write an equation for the line through π» and perpendicular to Μ Μ Μ Μ πΊπ». π»β² (π, π), πΊβ² (π β (βπ), βπ β ππ), π·β²(π β (βπ), π β ππ) (π β (βπ)) (π β (βπ)) + (βπ β ππ)(π β ππ) = π πππ + πππ = πππ β πππ ππ ππ π= π+ ππ π Closing (2 minutes) Describe the characteristics of a normal segment. A line segment with one endpoint on a line and perpendicular to the line is called a normal segment to the line. Every equation of a line through a given point (π, π) has the form π¨(π β π) + π©(π β π) = π. Explain how the values of π¨ and π© are obtained. π¨ is the value of the abscissa of the image of the endpoint of the normal segment that does not lie on the line after the figure has been translated so that the image of the endpoint that does lie on the line, the point of perpendicularity, is at the origin. π© is the value of the ordinate of the image of the endpoint of the normal segment that does not lie on the line after the figure has been translated so that the image of the endpoint that does lie on the line, the point of perpendicularity, is at the origin. Exit Ticket (5 minutes) Lesson 7: Equations for Lines Using Normal Segments This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M4-TE-1.3.0-09.2015 77 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 7 NYS COMMON CORE MATHEMATICS CURRICULUM M4 GEOMETRY Name Date Lesson 7: Equations for Lines Using Normal Segments Exit Ticket Given π΄(β5, β3), π΅(β1, 6), and πΆ(π₯, π¦): a. What are the coordinates of the translated points if π΅ moves to the origin? b. Write the condition for perpendicularity equation. c. Write the equation for the normal line in slope-intercept form. Lesson 7: Equations for Lines Using Normal Segments This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M4-TE-1.3.0-09.2015 78 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 7 M4 GEOMETRY Exit Ticket Sample Solutions Given π¨(βπ, βπ), π©(βπ, π), and πͺ(π, π): a. What are the coordinates of the translated points if π© moves to the origin? π¨β²(βπ, βπ), π©β²(π, π), πͺβ²(π + π, π β π) b. Write the condition for perpendicularity equation. βπ(π + π) β π(π β π) = π c. Write the equation for the normal line in slope-intercept form. π ππ π=β π+ π π Problem Set Sample Solutions 1. Given points πͺ(βπ, π) and π«(π, π): a. Μ Μ Μ Μ . Write the equation of the line through πͺ and perpendicular to πͺπ« π = βπ b. Write the equation of the line through π« and perpendicular to Μ Μ Μ Μ πͺπ«. π=π 2. Given points π΅(π, π) and π΄(π, βπ): a. Μ Μ Μ Μ Μ . Write the equation of the line through π΄ and perpendicular to π΄π΅ π = βπ b. Write the equation of the line through π΅ and perpendicular to Μ Μ Μ Μ Μ π΄π΅. π=π 3. The equation of a line is given by the equation π(π β π) + π(π + π) = π. a. What are the coordinates of the image of the endpoint of the normal segment that does not lie on the line? Explain your answer. (π, π) because π¨(π β π) + π©(π + π) = π is the original formula, and (π¨, π©) are the coordinates of the image of the endpoint of the normal segment not on the line. b. What translation occurred to move the point of perpendicularity to the origin? β©βπ, πβͺ, or left π units up π units Lesson 7: Equations for Lines Using Normal Segments This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M4-TE-1.3.0-09.2015 79 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 7 NYS COMMON CORE MATHEMATICS CURRICULUM M4 GEOMETRY c. What were the coordinates of the original point of perpendicularity? Explain your answer. (π, βπ) because the translation of β©βπ, πβͺ was required to move the point of perpendicularity to the origin. d. What were the endpoints of the original normal segment? π= πβπ and π = ππ π = π β (βπ) π =π The endpoints of a normal segment to the given line are π¨(π, βπ) and π©(ππ, π). 4. A coach is laying out lanes for a race. The lanes are perpendicular to a segment of the track such that one endpoint of the segment is (π, ππ), and the other is (ππ, ππ). What are the equations of the lines through the endpoints? π πππ π π= β π+ ; π = β π + ππ π ππ π Lesson 7: Equations for Lines Using Normal Segments This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M4-TE-1.3.0-09.2015 80 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.