# Find the Distance between

```Focus
and leave a trail. -- Ralph Waldo Emerson
Perpendiculars and Distance
Chapter 3
Section 6
Find the Distance between:
A point and a line, 2 parallel lines
Distance, point and a line
Draw the shortest distance between line q and
M
point M
q
You Draw Distance
Draw a segment that represents the distance
from point P to AB. P
A
B
Draw a segment that represents the distance
from point Q to RS.
S
Q
R
Distance
Distance on the coordinate plane always
involves 2 points.
Distance between. . .
1) 2 points
2) point and a line
3) 2 lines

Distance from a point to a line
t
Construct a line
perpendicular to line s
through V(1, 5) not on s.
Then find the distance from
V to s.
1. What do we know about perpendicular Lines?
slopes are “twice opposite”
2. Find the slope of line s then our new slope
ms = 1, mt = -1
??
Distance from a point to a line
t
3. Find the Distance
between the point and the
line.
Note: Line t appears to
intersect line s at (-2, 2)
Distance Between Parallel Lines
M
2.5 cm
s
q
Distance between 2 lines
Find the distance between the parallel lines
a and b whose equations are
y = 2x + 3 and y = 2x –3
You will need to solve a system of equations to find the
endpoints of a segment that is perpendicular to both a
and b. The slope of lines a and b is 2.
y = 2x + 3 and y = 2x –3
1. Wisely choose any point
(0, 3)
on the line
2. Slope of line p ? mp = -1/2
3. What do we know?
point and slope
Point-slope form
Simplify.
What do we need for Distance?

What do we have now?
p
a
b
y = 2x + 3
y = 2x – 3
p and a
intersect at
(0, 3)
p and b
intersect at
???
Find the intersection and we
can plug it into our formula
Find the intersection of 2 lines

Where do they intersect? At the point they share
Group like terms on each side.
Simplify on each side.
Divide each side by
.
Substitute 2.4 for x in the
equation for p.
The point of intersection is (2.4, 1.8).
What do we have now?

Our 2 points for distance
formula
(0, 3)
Distance Formula
x2 = 2.4, x1 = 0,
y2 = 1.8, y1 = 3
(2.4, 1.8)
Distance and Parallel Lines
Find the distance between the parallel lines a and b
whose equations are
and
2.85 units
Theorum 3.9
Homework

p. 185-186 #5-10, 12-18(evens) 22, 29-31
(15 problems)
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