1. 3.5 Parallel and PerpendicularLines in the Coordinate Plane
Geometry
3.5 Parallel and PerpendicularLines in the Coordinate Plane
EQ: How do find the equation of lines || and perp to other lines?

2. Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane
Topic/Objective
Find the slope of lines on the coordinate plane.
Determine if two lines are parallel or perpendicular.
Write the equation of parallel and perpendicular lines.
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

3. Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane
Review: Slope
Slope =
Rise
Run
Run = 6
(3, 3)
Rise =4
(-3, -1)
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

4. Geometry 3.5 Parallel & Perpendicular 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.
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

5. Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane
Another Example
Slope =
Rise
Run
Run = -3
(-1, 3)
Rise =3
(2, 0)
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

6. We can also use the formula.
Given two points
and
The slope is
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

7. Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane
Example
Find the slope of the line that passes through (9, 12) and (6, -3).
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

8. Theorem 3.13 Slopes of parallel lines
Parallel lines have the same slope.
We write: m1 = m2
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

9. Theorem 3.14 Slopes of Perp lines
Two lines are perpendicular iff the product of their slopes is –1.
Algebraically: m1 • m2 = –1
A vertical and a horizontal line are perpendicular.
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

10. Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane
Example m1 2 1 -1 2
m1  m2 m2
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

11. 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!
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

12. Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane
Equation of a Line
Slope-Intercept form: y = mx + b
m is the slope
b is the y-intercept
The y-intercept is the value of y where the line crosses the y-axis.
The y-intercept has the coordinate (0, b)
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

13. Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane
Slope-Intercept Form
To be able to write an equation in slope-intercept form, you must know two things:
The slope of the line, m.
The y-intercept, b.
y=mx + b
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

14. Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane
Example
Write the equation of a line with slope of 3 and y-intercept of 8.
y = mx + b
y = 3x + 8
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

15. Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane
Example
Write the equation of a line parallel to y = 5x + 10 that has a y-intercept of –6.
Think: Parallel Lines = Same Slope
y = mx + b
m = 5
b = –6
y = 5x – 6
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

16. Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane
Another Example
Write the equation of a line parallel to y = x that has a y-intercept of 2.
Think: Parallel Lines = Same Slope
y = mx + b
m = 1
b = 2
y = x+2
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

17. Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane
Example
Write the equation of a line with a slope of –2 that passes through (4, 1).
y = mx + b
Now solve for b.
1 = -2(4) + b
1 = -8 + b
9 = b
y = -2x + 9
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

18. Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane
You try it…
Write the equation of a line that has a slope of 4 and passes through (1, –3).
Solution:
y = mx + b. m = ?
y = 4x + b. x = ? y = ?
–3 = 4(1) + b
–7 = b
y = 4x – 7
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

19. Write the equation of a line with no slope that passes through (3, 5).
x = 3
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

20. Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane
Perpendicular slope
6 = 18 + b
-12 = b
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

21. Find the distance from the point (1,0) to the line y = -x + 3
Step I: Find equation line perp to y = -x + 3 passing through (1,0)
m = = 1(1) + b
-1 = b
y = 1x - 1
Step II: Now find the point of intersection of the two lines by solving a system of equations.
y = -x + 3
y = x – 1
(2,1)
2y = 2
y = 1
1 = x – 1
x = 2
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

22. Find the distance from the point (1,0) to the line y = -x + 3
Step III
Now find the distance between (1,0) and (2,1)
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

23. Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane
Summary
Slope measures the steepness of a line.
Slope is the Rise/Run.
Slope intercept form is y = mx + b.
The y-intercept is (0, b).
Parallel lines have the same slope.
Perpendicular slopes have a product of -1
February 23, 2019
Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane