1. 3.7 Perpendicular Lines in the Coordinate Plane
Geometry
3.7 Perpendicular Lines in the Coordinate Plane

2. 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.
April 20, 2017
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

3. 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).
April 20, 2017
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

4. 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
April 20, 2017
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

5. 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.
April 20, 2017
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

6. 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.
April 20, 2017
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

7. Geometry 3.7 Perpendicular Lines in the Coordinate Plane
Example m1 2 1 -1 2
m1  m2 m2
April 20, 2017
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

8. 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!
April 20, 2017
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

9. lines are perpendicular.
If the product of the
slopes is -1, then the
lines are perpendicular.
April 20, 2017
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

10. 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.
April 20, 2017
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

11. Another way to think of it:
Two lines are perpendicular if one slope is the negative reciprocal of the other.
April 20, 2017
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

12. 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.
April 20, 2017
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

13. More challenging problem
These equations are in General Form
Ax + By = C
Slope is always:
April 20, 2017
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

14. 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
April 20, 2017
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

15. 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.
April 20, 2017
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

16. 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.
April 20, 2017
Geometry 3.7 Perpendicular Lines in the Coordinate Plane

17. Geometry 3.7 Perpendicular Lines in the Coordinate Plane
Homework
April 20, 2017
Geometry 3.7 Perpendicular Lines in the Coordinate Plane