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Lesson 3-1 Parallel Lines and Transversals
Lesson 3-2 Angles and Parallel Lines
Lesson 3-3 Slopes of Lines
Lesson 3-4 Equations of Lines
Lesson 3-5 Proving Lines Parallel
Lesson 3-6 Perpendiculars and Distance
Example 1 Distance from a Point to a Line
Example 2 Construct a Perpendicular Segment
Example 3 Distance Between Lines
OBJECTIVE: To find the distance between a point
and a line and between parallel lines (2.9.8E)
(M8.C.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,
Draw the segment that represents the distance from
Answer:
Construct a line perpendicular to line s through V(1, 5)
not on s. Then find the distance from V to s.
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.
Put the compass at point A and draw an arc below line s.
(Hint: Any compass setting greater than
will work.)
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.
Draw
.
and s. Use the slopes of
lines are perpendicular.
and s to verify that the
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.
Construct a line perpendicular to line m through
Q(–4, –1) not on m. Then find the distance from
Q to m.
Answer:
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.
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.
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.
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).
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
Find the distance between the parallel lines a and b
whose equations are
respectively.
Answer:
and
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