# Perpendiculars and Distance ```Perpendiculars and Distance
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You proved that two lines are parallel using angle
relationships.
• Find the distance between a point and a line.
• Find the distance between parallel lines.
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Construct Distance From Point to a Line
CONSTRUCTION A certain roof
truss is designed so that the
center post extends from the
peak of the roof (point A) to the
main beam. Construct and
name the segment whose
length represents the shortest
length of wood that will be
needed to connect the peak
of the roof to the main beam.
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. Locate points R and S on the main beam equidistant
from point A.
Locate a second point not on
the beam equidistant from R
and S. Construct AB so that
AB is perpendicular to the beam.
Answer: The measure of 𝐴𝐵 represents the
shortest length of wood needed to connect the
peak of the roof to the main beam.
KITES Which segment represents the shortest
distance from point A to DB?
B. AB
C. CX
D. AX
COORDINATE GEOMETRY Line s
contains points at (0, 0) and (–5, 5).
Find the distance between line s and
point V(1, 5).
Step 1 Find the slope of line s.
(–5, 5)
V(1, 5)
(0, 0)
Begin by finding the slope of the
line through points (0, 0) and
(–5, 5).
Then write the equation of this line by using the point
(0, 0) on the line.
Slope-intercept form
m = –1, (x1, y1) = (0, 0)
Simplify.
The equation of line s is y = –x.
Step 2
Write an equation of the line t perpendicular
to line s through V(1, 5).
Since the slope of line s is –1, the slope of line t is 1.
Write the equation for line t through V(1, 5) with a slope
of 1.
Slope-intercept form
m = 1, (x1, y1) = (1, 5)
Simplify.
Subtract 1 from each side.
The equation of line t is y = x + 4.
Step 3
line s:
Solve the system of equations to determine
the point of intersection.
y = –x
line t: (+) y = x + 4
2y = 4
y= 2
Divide each side by 2.
Solve for x.
2 = –x
–2 = x
Substitute 2 for y in the first
equation.
Divide each side by –1.
The point of intersection is (–2, 2). Let this point be Z.
Step 4
Use the Distance Formula to determine the
distance between Z(–2, 2) and V(1, 5).
Distance formula
Substitution
Simplify.
Answer: The distance between the point and the line is
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By definition, parallel lines do not intersect. An alternate
definition states that two lines in a plane are parallel if
they are everywhere equidistant.
Equidistant means that the distance between two lines
measured along a perpendicular line to the lines is always
the same.
Steps to follow…
1. Find the slope
2. Write the equation of the line.
3. Write the equation of a line perpendicular
through the point given.
4. Take equations from #2 and #3 and solve for
x and y using elimination.
5. Use the distance formula to find the distance
between the given point and the point you
found in #4.
𝑑 = 𝑥2 − 𝑥1 2 + 𝑦2 − 𝑦1 2
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Find the distance between the parallel lines a and
b whose equations are y = 2x + 3 and y = 2x – 1,
respectively.
You will need to solve a system
a b
of equations to find
the endpoints of a segment
that is perpendicular to both
a and b. From their equations,
we know that the slope of line
a and line b is 2.
p
Sketch line p through the
y-intercept of line b, (0, –1),
perpendicular to lines a and b.
Write an equation for line p. The slope of p is the
opposite reciprocal of
Use the y-intercept of line b, (0, –1), as one of the
endpoints of the perpendicular segment.
Point-slope form
Simplify.
Subtract 1 from each side.
Use a system of equations to determine the point of
intersection of the lines a and p.
Substitute 2x + 3 for y in the
second equation.
Group like terms on
each side.
Simplify on each side.
Multiply each side by
Substitute
equation for p.
.
for x in the
Simplify.
Step 3
Use the Distance Formula to determine the distance
between (0, –1) and (–1.6, –0.2).
Distance Formula
x2 = –1.6, x1 = 0,
y2 = –0.2, y1 = –1
The distance between the lines is about 1.79
units.
Steps to follow…
1. Find the slope
2. Write the equation of the line.
3. Write the equation of a line perpendicular
through the point given.
4. Take equations from #2 and #3 and solve for
x and y using elimination.
5. Use the distance formula to find the
distance between the given point and the
point you found in #4.
𝒅 = 𝒙𝟐 − 𝒙𝟏 𝟐 + 𝒚𝟐 − 𝒚𝟏 𝟐
3-6 Assignment
Page 221, 13, 14, 15, 17, 21, 23
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