1. Lesson 3 – 6 Perpendicular and Distance
Geometry
Lesson 3 – 6
Perpendicular and Distance
Objective:
Find the distance between a point and a line.
Find the distance between parallel lines.

2. Distance from a point to a line?
Don’t copy
Distance from a point to a line?
The distance between a line and a point not on the line is the length of the segment perpendicular to the line from the point.
Perpendicular postulate
If given a line and a point not on the line, then there exits exactly one line through the point that is perpendicular to the given line.

3. Line l contains points (-5, 3) and (4, -6)
Line l contains points (-5, 3) and (4, -6). Find the distance between l and point (2, 4).
Step 1: write the equation of given line
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m = -1
y = -x - 2
Need the shortest distance
from P to the line so we need the
Perpendicular line from P.
Step 2: Write an equation
perpendicular and through
the given point.
m = 1
y = x + 2
Cont…

4. Copy
Step 3: Find the point of intersection by solving the system of equations.
y = -x – 2
y = x + 2
2y = 0
y = 0
0 = x + 2
-2 = x
If the point of intersection can
be found using the graph that is
fine, but if not system of
equations must be used.
(-2, 0) Q
Cont…

5. I want both for your answer!
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Step 4: Find the distance from the given point to the point of intersection.
Distance between (2, 4) and (-2, 0)
Simplify!
I want both for
your answer!

6. Try
Line l contains points (1, 2) and (5, 4). Find the distance between l and point P (1, 7).
Step 1: write the equation of the line
m = ½
2 = (1/2)(1) + b
b = 3/2
y = ½ x + 3/2
Step 2: Write an equation perpendicular and through the given point.
7 = -2(1) + b
b = 9
y = -2x + 9
Cont…

7. Step 3: Find the point of intersection by solving the system of equations.
y = ½ x + 3/2
y = -2x + 9
y = -2(3) + 9 =
= 3
(3, 3)
x = 3
Cont…

8. Step 4: Find the distance from the given point to the point of intersection.
(3, 3) (1, 7)

9. Step 1: pick a point (any point) on either line
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Find the distance between the parallel lines l and m with equations y = 2x + 1 and y = 2x – 3, respectively.
Step 1: pick a point (any point) on either line
HINT – pick an easy point like a y-intercept!
(0, 1) (y-intercept of first equation)
Keep track of this point you will need it later!
Step 2: Write an equation perpendicular and through the point from step 1.
M = -1/2 (perpendicular) thru (0, 1)
1 = (-1/2)(0) + b
Respectively means the equations are in the same order as the named lines.

10. Step 3: Use system of equations to find point of intersection
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Step 3: Use system of equations to find point of intersection
2 equations you use
Perpendicular equation
The equation that you didn’t use y-intercept
y = 2x – 3

11. Step 4:Find the distance between the two points
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Step 4:Find the distance between the two points
Point from step 1 and the point of intersection.
(0, 1)

12. Homework
Pg – 32 EOE