1. Parallel & Perpendicular Lines
Unit 1
Parallel & Perpendicular Lines

2. Section 1
Slope-Intercept Review

3. Slope-Intercept Form of a Line
y = mx + b
x and y are variables
m and b are numbers
m is the slope
“Rise over run”
b is the y-intercept
Point where the line crosses the y-axis

4. Example
y = 2x – 1
Slope: m = 2 =
y-intercept: b = -1 2 1

5. Example Set 1
Identify the slope and y-intercept of each line, given its equation:
y = ½x – 3
y = 3x
y = -5 – 2x

6. Example 2
Given equation 2x + 3y = 7, can we immediately find the slope and y-intercept?
No! We must first put it in slope-intercept form!
Get the y by itself:
2x + 3y = 7
-2x x
3y = -2x + 7 3
y = -2/3x + 7/3
Slope: -2/3
y-intercept: 7/3

7. Section 2
Identifying Parallel/Perpendicular Lines

8. Parallel vs. Perpendicular
Parallel: Two lines in the same plane that never intersect
Parallel Lines have equal slopes
Perpendicular: Two lines that intersect to form a 90° angle
Perpendicular lines have opposite reciprocal slopes
Opposite: different signs (+/-)
Reciprocal: flip the fraction

9. Example Set 3 Find the opposite reciprocal of the following numbers:
3    - -4 1 3 1 1 3 1 3 2 5

10. Parallel, Perpendicular, or Neither?

11. Section 3
Finding the Equation of a Parallel/Perpendicular Line through a Point

12. Announcements EXTENSION: Homework Packet due Monday
Unit 1 Test POSTPONED until Monday

13. Finding the Equation of a Parallel Line
Sample Problem:
Write the equation of a line parallel to the line
y = -2x + 3 that passes through the point (2,1).
Think back: What do you know about the slopes of parallel lines?

14. Finding the Equation… continued
Write the equation of a line parallel to the line
y = -2x + 3 that passes through the point (2,1).
Find the slope m of a parallel line.
Plug slope into y = mx + b.
Plug x and y-values into equation from step 2.
Simplify, solve for b.
Rewrite equation using new m and b values.
m = -2
y = -2x + b
1 = -2(2) + b  1 = -4 + b
5 = b
y = -2x + 5

15. Finding the Equation of a Perpendicular Line
PREDICT: How might the steps be different if we’re finding the equation of a perpendicular line through a point?
HINT: What do we know about the slopes of perpendicular lines?

16. Finding the Equation… continued
Write the equation of a line perpendicular to the line
y = -2x + 3 that passes through the point (2,1).
Find the slope m of a perpendicular line.
Plug slope into y = mx + b.
Plug x and y-values into equation from step 2.
Simplify, solve for b.
Rewrite equation using new m and b values.
Opposite reciprocal! 1 2
m = 1 2
y = x + b 1 2
1 = (2) + b  1 = 1 + b
-1_ -1____
0 = b
y = x 1 2

17. Practice I will pick people to come up to the board for each problem!
Find the equation of a line parallel to the line
-3x + y – 2 = 4 that passes through the point (-2,-4).
y = 3x + 2
Find the equation of a line perpendicular to the line -x – 2y = 6 that passes through the point (4,-1).
y = -2x + 7
I will pick people to come up to the board for each problem!

18. Wrap Up
Exit Slip
Remember, homework packet and test now for MONDAY the 27th

19. Section 4
Midpoint Formula

20. Midpoint What is the midpoint of a line? Midpoint Formula:
Point on the line equidistant from the two endpoin
Midpoint Formula:
Notice it’s just the average of the two x-values and the average of the two y-values!

21. Example
What is the midpoint of the line segment with endpoints at A(3, -4) and B(5, -1)?

22. Practice
Find the midpoint of line segment AB with endpoints A(4, -6) and B(-4, 2).
Find the midpoint of line segment CD with endpoints C(0, -8) and D(3, 0).
Find the midpoint of line segment XY with endpoints X(-3, -7) and Y(-1, 1)
Find the midpoint of line segment LN with endpoints L(12, -7) and N(-5, -2)

23. Finding the Other Endpoint
How do we find the other endpoint if we know the midpoint and first endpoint?
Example: Find the endpoint B of line segment AB, with endpoint A(0,-5) and midpoint M(2,-3).
Try coming up with the answer by graphing the endpoint and the midpoint.
How many spaces up and to the right should the other endpoint be?

24. Math Challenge
Can you come up with a way to find the other endpoint algebraically (without graphing)?
Example: Find the endpoint B of line segment AB, with endpoint A(0,-5) and midpoint M(2,-3).

25. Practice
M is the midpoint of QR with Q(-3, 5) and M(7, -9). Find the coordinates of R.
D is the midpoint of CE with E(-3, -2) and D(5, 1). Find the coordinates of C.
M is the midpoint of LN with L(0, 0) and M(-2, -8). Find the coordinates of N.

26. Wrap Up
Exit Slip
Remember, homework packet and test now for MONDAY the 27th

27. Section 5
Distance Formula

28. How do we find the distance between two points?
Example: Line segment AB has endpoints A(5, 4) and B(3,-2). Find the length of AB.
Hint: Can you figure it out by graphing AB?

29. Finding the distance continued
Horizontal distance = 8
Vertical distance = 6
Pythagorean Theorem:
a2 + b2 = c2
= 100
d = √100 = 10

30. Distance Formula
We can also plug A(-3, -2) and B(5, 4) into this formula:
Example:

31. Practice
The endpoints of RT are R(-1,-2) and T(5, 6). What is the length of RT?
The endpoints of AB are A(0, 7) and B(-3, 11). Find the length of AB.
Tanya runs diagonally across a rectangular field that has a length of 40 yards and a width of 30 yards, as shown in the diagram below. What is the length of the diagonal, in yards, that Tanya runs?

32. Wrap Up
Exit Slip
Remember, homework packet and test now for MONDAY the 27th

33. Section 6
Finding the Perpendicular Bisector

34. What does it mean to bisect something?
Bisect: to split in half
PREDICT: What is a perpendicular bisector?
Line that is perpendicular to a line segment and splits it in half

35. Finding the Equation of a Perpendicular Bisector from Two Endpoints
Example: Find the equation of the perpendicular bisector of the line segment with endpoints A(2, 3) and B(-2, -5).
Similar to Monday’s lesson with finding the equation of a perpendicular line, with two differences:
You have to calculate the slope using the slope equation
You must calculate the midpoint and plug it in

36. Equation of a Perpendicular Bisector continued
Find the equation of the perpendicular bisector of the line segment with endpoints A(2, 3) and B(-2, -5).
Calculate the slope using the slope formula.
Find the opposite reciprocal.
Plug it into the equation y = mx + b.

37. Equation of a Perpendicular Bisector continued
A(2,3) and B(-2,-5)
Find the midpoint of AB.
Plug coordinates of midpoint into equation.
Solve for b.
Rewrite Equation with m and b.