1. Parallel & Perpendicular Lines in the Coordinate Plane
Geometry
Parallel & Perpendicular
Lines in the Coordinate
Plane

2. Geometry 3.6 Parallel Lines in the Coordinate Plane
Goals
Find the slope of lines on the coordinate plane.
Determine if two lines are parallel.
December 10, 2018
Geometry 3.6 Parallel Lines in the Coordinate Plane

3. Geometry 3.6 Parallel Lines in the Coordinate Plane
Review: Slope
Slope =
Rise
Run
Run = 6
(3, 3)
Rise =4
(-3, -1)
December 10, 2018
Geometry 3.6 Parallel Lines in the Coordinate Plane

4. Geometry 3.6 Parallel Lines in the Coordinate Plane
Reminder
Lines with a positive slope rise to the right.
Lines with a negative slope rise to the left.
Lines with zero slope are horizontal.
Lines with no slope are vertical.
December 10, 2018
Geometry 3.6 Parallel Lines in the Coordinate Plane

5. Geometry 3.6 Parallel Lines in the Coordinate Plane
Another Example
Slope =
Rise
Run
Run = -3
(-1, 3)
Rise =3
(2, 0)
December 10, 2018
Geometry 3.6 Parallel Lines in the Coordinate Plane

6. We can also use the formula.
Given two points
and
The slope is
December 10, 2018
Geometry 3.6 Parallel Lines in the Coordinate Plane

7. Geometry 3.6 Parallel Lines in the Coordinate Plane
Example
Find the slope of the line that passes through (9, 12) and (6, -3).
December 10, 2018
Geometry 3.6 Parallel Lines in the Coordinate Plane

8. Geometry 3.6 Parallel Lines in the Coordinate Plane
Postulate 17
Parallel lines have the same slope.
We write:
m1 = m2
December 10, 2018
Geometry 3.6 Parallel Lines in the Coordinate Plane

9. Geometry 3.6 Parallel Lines in the Coordinate Plane
Summary
Slope measures the steepness of a line.
Slope is the Rise/Run.
Parallel lines have the same slope.
December 10, 2018
Geometry 3.6 Parallel Lines in the Coordinate Plane

10. Perpendicular Lines in the Coordinate Plane
Geometry
Perpendicular Lines in the Coordinate Plane

11. Geometry 3.7 Perpendicular Lines in the Coordinate Plane
Goals
Use slope to identify perpendicular lines in a coordinate plane.
Write equations of perpendicular lines.
December 10, 2018
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

12. Geometry 3.7 Perpendicular Lines in the Coordinate Plane
Review
Lines are parallel if they have the same slope.
Slope is rise/run.
Lines with a positive slope rise to the right.
Lines with a negative slope rise to the left.
Horizontal Lines: slope = 0.
Vertical Lines: slope is undefined (or none).
December 10, 2018
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

13. Geometry 3.7 Perpendicular Lines in the Coordinate Plane
Problem Slopes
Find the slope of the line containing (4, 6) and (2, 6).
Do it graphically:
(2, 6)
(4, 6)
Horizontal Lines have the form y = c.
y = 6
December 10, 2018
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

14. Geometry 3.7 Perpendicular Lines in the Coordinate Plane
Problem Slope
Find the slope of the line containing (4, 6) and (4, 3).
Do it graphically:
(4, 6)
(4, 3)
undefined (no SlopE)
x = 4
vertical Lines have the form x = c.
December 10, 2018
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

15. Geometry 3.7 Perpendicular Lines in the Coordinate Plane
Postulate 18
Two lines are perpendicular iff the product of their slopes is –1.
Algebraically: m1 • m2 = –1
A vertical and a horizontal line are perpendicular.
December 10, 2018
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

16. Geometry 3.7 Perpendicular Lines in the Coordinate Plane
Example m1 2 1 -1 2
m1  m2 m2
December 10, 2018
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

17. You don’t need a picture.
Line A contains (2, 7) and (4, 13).
Line B contains (3, 0) and (6, -1).
Are the lines perpendicular?
YES!
December 10, 2018
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

18. lines are perpendicular.
If the product of the
slopes is -1, then the
lines are perpendicular.
December 10, 2018
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

19. Geometry 3.7 Perpendicular Lines in the Coordinate Plane
Exception
Slope of m1 is ?
 Undefined
Slope of m2 is ?
 Zero
m1  m2  –1.
But m1  m2! m1
(2, 2) m2
(-2, 1)
(3, 1)
(2, -1)
A vertical line and a horizontal line are perpendicular.
December 10, 2018
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

20. Another way to think of it:
Two lines are perpendicular if one slope is the negative reciprocal of the other.
December 10, 2018
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

21. Slope Intercept form review
y = mx + b
m is the slope
b is the y-intercept
The y-intercept is at (0, b)
Lines are parallel if they have the same slope. They are perpendicular if the product of their slopes is –1.
December 10, 2018
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

22. More challenging problem
These equations are in General Form
Ax + By = C
Slope is always:
December 10, 2018
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

23. Geometry 3.7 Perpendicular Lines in the Coordinate Plane
Why is this so?
Consider the equation: 8x – 4y = 12
Move the 8x: – 4y = – 8x + 12
Divide by –4: y = 2x – 3
Slope is? 2
Now use –A/B: -8/(-4) = 2
December 10, 2018
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

24. The slopes are negative reciprocals, so the lines are perpendicular.
For –3x + 2y = 2, slope is
For 2x + 3y = –2, slope is
The slopes are negative reciprocals, so the lines are perpendicular.
December 10, 2018
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

25. Geometry 3.7 Perpendicular Lines in the Coordinate Plane
In summary
Two lines are parallel if they have the same slope.
Two lines are perpendicular if the product of their slopes is –1.
General form is Ax + By = C and the slope in this form is –A/B.
December 10, 2018
Geometry 3.7 Perpendicular Lines in the Coordinate Plane