1. 4.3 Parallel and perpendicular lines

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

3. 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

4. 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) = − = −1 5
So, line a is parallel to line c
𝑎∥𝑐 is symbolic way to say

5. 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

6. 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

7. 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
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

8. 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= 𝑏
2=6+𝑏
−6−6
−4=𝑏
Parallel equation:
𝑦= 1 3 𝑥−4

9. Ex. 3 Perpendicular lines
Slopes of perpendicular lines are negative reciprocals of each other
For example:
m = 2 and m = − 1 2 ; m = 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?

10. 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

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

12. 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
𝑦= 3 2 𝑥−17