3.6 Perpendiculars and Distance
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Distance from a Point to a Line
The distance from a line to a point not on
the line is the length of the segment ┴ to
the line from the point.
l
A
Example 1:
Draw the segment that represents the distance from
Answer:
Since the distance from a line to a point not on the line is
the length of the segment perpendicular to the line from
the point,
Your Turn:
Turn to Pages 162 – 163
in your textbook and
complete #11 – 16.
Constructing a ┴ Segment

How do we construct a ┴ segment
accurately?
By using a compass.

How do we use a compass?
The next example will show us.
Example 2:
Construct a line perpendicular to line s through V(1, 5)
not on s. Then find the distance from V to s.
Example 2:
Graph line s and point V. Place the compass point at point
V. Make the setting wide enough so that when an arc is
drawn, it intersects s in two places. Label these points of
intersection A and B.
Example 2:
Put the compass at point A and draw an arc below line s.
(Hint: Any compass setting greater than
will work.)
Example 2:
Using the same compass setting, put the compass at
point B and draw an arc to intersect the one drawn in
step 2. Label the point of intersection Q.
Example 2:
Draw
.
and s. Use the slopes of
lines are perpendicular.
and s to verify that the
Example 2:
The segment constructed from point V(1, 5)
perpendicular to the line s, appears to intersect line s at
R(–2, 2). Use the Distance Formula to find the distance
between point V and line s.
Answer: The distance between V and s is about 4.24 units.
Your Turn:
Turn to Page 163
in your textbook and
complete #17 – 18.
Distance Between Parallel Lines

Two lines in a plane are || if they are
equidistant everywhere.

To verify if two lines are equidistant find the
distance between the two || lines by
calculating the distance between one of the
lines and any point on the other line.
Theorem 3.9
In a plane, if two lines are equidistant from
a third line, then the two lines are || to each
other.
Example 3:
Find the distance between the parallel lines a and b
whose equations are
and
respectively.
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.
Example 3:
First, write an equation of a line p perpendicular to a and b.
The slope of p is the opposite reciprocal of 2,
Use the y-intercept of line a, (0, 3), as one of the endpoints
of the perpendicular segment.
Point-slope form
Simplify.
Add 3 to each side.
Example 3:
Next, use a system of equations to determine the point
of intersection of line b and p.
Substitute 2x–3 for y in
the second equation.
Example 3:
Group like terms on each side.
Simplify on each side.
Substitute 2.4 for x in the equation
for p.
The point of intersection is (2.4, 1.8).
Example 3:
Then, use the Distance Formula to determine the distance
between (0, 3) and (2.4, 1.8).
Distance Formula
Answer: The distance between the lines is
2.7 units.
or about
Your Turn:
Turn to Page 163
in your textbook and
complete #19 – 22.
Pre-AP: Add #24