4.3 Parallel and perpendicular lines

What we will learn Identify and write equations of parallel lines Identify and write equations of perpendicular lines

Needed vocab Parallel lines: two lines in the same plane that never intersect Perpendicular lines: two lines in the same plane that intersect to form right angles

Ex. 1 identifying parallel lines Parallel lines have the same slope Find slope of each line Use slope formula 𝑚= 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 𝑚 𝑎 = 2−3 1−(−4) = 2−3 1+4 = −1 5 𝑚 𝑏 = −1−0 1− −3 = −1−0 1+3 = −1 4 𝑚 𝑐 = −5−(−4) 2−(−3) = −5+4 2+3 = −1 5 So, line a is parallel to line c 𝑎∥𝑐 is symbolic way to say

Your practice Line a goes through (-5,3) and (-6,-1). Line b goes through (3,-2) and (2,-7). Are a and b parallel? 𝑚 𝑎 = −1−3 −6−(−5) = −1−3 −6+5 = −4 −1 =4 𝑚 𝑏 = −7−(−2) 2−3 = −7+2 2−3 = −5 −1 =5 no

Ex. 2 write equation of parallel lines Write equation of a line through (5,-4) and parallel to 𝑦=2𝑥+3. New slope: m = 2 Finding b: 𝑦=𝑚𝑥+𝑏 −4=2 5 +𝑏 −4=10+𝑏 −10 −10 −14=𝑏 Write parallel equation: 𝑦=2𝑥−14 Steps 1. put given equation into 𝑦=𝑚𝑥+𝑏 form 2. find slope of new line from old m = slope Will be the same as the old slope 3. find b Plug in m, x, and y into 𝑦=𝑚𝑥+𝑏 and solve for b x and y come from given point in problem 4. write 𝑦=𝑚𝑥+𝑏 by plugging in new m and b

Ex. 2 Continued Write equation through (-4,2) and parallel to −𝑥+4𝑦=4. Get into y = mx + b form: −𝑥+4𝑦=4 +𝑥 +𝑥 4𝑦=𝑥+4 4𝑦 4 = 𝑥 4 + 4 4 𝑦= 1 4 𝑥+1 New slope: m = 1 4 Finding b: 2= 1 4 −4 +𝑏 2=−1+𝑏 +1 +1 3=𝑏 Parallel equation: 𝑦= 1 4 𝑥+3

Your practice Write an equation of a line through (18,2) and parallel to 3y – x = -12. 3𝑦−𝑥=−12 +𝑥 +𝑥 3𝑦=𝑥−12 3𝑦 3 = 𝑥 3 − 12 3 𝑦= 1 3 𝑥−4 New slope: m = 1 3 Finding b: 2= 1 3 18 +𝑏 2=6+𝑏 −6−6 −4=𝑏 Parallel equation: 𝑦= 1 3 𝑥−4

Ex. 3 Perpendicular lines Slopes of perpendicular lines are negative reciprocals of each other For example: m = 2 and m = − 1 2 ; m = 3 2 and m = −2 3 Find slopes 𝑚 𝑎 = 3−1 0−(−2) = 3−1 0+2 = 2 2 =1 𝑚 𝑏 = 4−1 6−4 = 3 2 𝑚 𝑐 = 1−3 4−1 = −2 3 So b is perpendicular to c 𝑏⊥𝑐 is symbolic way to write Line a (-2,1) and (0,3) Line b (4,1) and (6,4) Line c (1,3) and (4,1) Which are perpendicular?

Ex. 4 Writing equations of perpendicular lines Write equation through (-3,1) and perpendicular to 𝑦= 1 2 𝑥+3. New Slope: 𝑚=−2 Finding b: 1=−2 −3 +𝑏 1=6+𝑏 −6 −6 −5=𝑏 New Perpendicular equation: 𝑦=−2𝑥−5 Steps 1. put given equation into 𝑦=𝑚𝑥+𝑏 form 2. find slope of new line from old m = slope Will be negative reciprocal of the old slope 3. find b Plug in m, x, and y into 𝑦=𝑚𝑥+𝑏 and solve for b x and y come from given point in problem 4. write 𝑦=𝑚𝑥+𝑏 by plugging in new m and b

Your practice Write equation through (8,1) perpendicular to 2𝑦+4𝑥=12. −4𝑥 −4𝑥 2𝑦=−4𝑥+12 2𝑦 2 = −4𝑥 2 + 12 2 𝑦=−2𝑥+6 New slope: 𝑚= 1 2 Finding b: 1= 1 2 8 +𝑏 1=4+𝑏 −4−4 −3=𝑏 New Equation: 𝑦= 1 2 𝑥−3

Ex. 5 Story Problems Slope of shoreline first: The position of a helicopter search and rescue crew is shown in the graph. The shortest flight path to the shoreline is one that is perpendicular to the shoreline. Write an equation that represents this path. Slope of shoreline first: m = −2 3 Slope of perpendicular line: m = 3 2 Use point - slope with point of helicopter 𝑦−4= 3 2 𝑥−14 𝑦−4= 3 2 𝑥−21 +4 +4 𝑦= 3 2 𝑥−17