Chapter 3.6 Perpendiculars and Distance
Check.4.8 , Apply properties and theorems about angles associated with
parallel and perpendicular lines to solve problems.
CLE 3108.3.1 Use analytic geometry tools to explore geometric problems
involving parallel and perpendicular lines, circles, and special points of
polygons.
Given 9  13, which segments are parallel?
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A. AB || CD
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B. FG || HI
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C. CD || FG
D. none
Two lines in the same plane do not intersect.
Which term best describes the relationship
between the lines?
A. parallel
B. perpendicular
C. skew
D. transversal
Question
• What is the quickest way to get from this classroom to the
cafeteria?
• If you are on the playing field or court, what is the quickest
route to leave the field?
Construction of Parallel Line through a Point
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Draw a Line, two points M, N
Draw point P not on the line.
Draw PM
Copy PMN so P is the vertex of
the new angle
• Know that PMN  RPQ then
MN  PQ
R
P Q
N
M
Construction of a Perpendicular Segment
• Use Graph Paper. Line l goes
through points (-6, -9) and (0,-1)
• Construct a line perpendicular to
line l through P (-7, -2)
l
P
B
• Put the compass at P and draw
an arc intersecting l in two
places, label these A and B
A
Q
• Put the compass at A and drawn
an arc below line l (must be ½
length of AB)
• Draw PQ. Check the
• Using same setting put the
slope to verify
compass at point B and draw
perpendicularity
and arc intersecting the previous
one.
• Label the intersection Q
Construction of a Perpendicular Segment
• Use the distance formula to
calculate the distance between P
and line l .
• (-7, -2) P
• (-3, -5) intersection
l
P
B
A
Q
Construction of a Perpendicular Segment
• Use Graph Paper.
• Construct a line perpendicular to
line s (y = -x) through V (1,5)
• Find the Distance from V to s
• Distance is 18
V
Distance between Point and a Line
• The shortest distance between a point and a line is
the length of the segment perpendicular to the line
from that point
C
A
B
Distance from Point to a line
C
A
B
Distance between Parallel Lines
• Two parallel lines are equidistant from each other. The
distance between two parallel lines is the distance between
one of the lines and any point on the other line
l
m
n
• Theorem – In a plane, if two lines are
equidistance from a third, then the two lines
are parallel to each other
Distance between two parallel
lines
3
8
Distance between two lines
• Find the distance between the parallel lines l and m whose
equations are y = -1/3x -3 and y = -1/3x+ 1/3 respectively.
1. Draw perpendicular line
2. Find the point where the two lines
cross.
3. Calculate Distance between points
• Solve system of equations
• Write equation for line perpendicular
passing through line
• m = 3, point (0, -3)
• y – y1 = m (x – x1)
• y –(-3)=3(x – 0)
• y +3 = 3x
• y = 3x - 3
Find the intersection of the two lines or the solution to the system
of equations
y = 3x – 3 and y = -1/3x+ 1/3
Substitute for y
3x – 3 = -1/3x+ 1/3
+1/3 x + 3 = +1/3x +3
3 1/3x = 3 1/3
x=1
Substitute x into either equation
y = 3(1) – 3
y=0
Calculate distance between
the two points
(0, -3) and (1, 0)
Find the distance between the parallel lines
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y = 2x +3 and y = 2x – 3
m = -1/2 point (0, -3)
y – (-3) = -1/2(x – 0)
y + 3 = -1/2x
y = -1/2x -3 and y = 2x + 3
-1/2x -3 = 2x + 3
+1/2x - 3 + 1/2x - 3
-6 = 2.5x
x= -2.4 , y = 2(-2.4)+3=-1.8
(0, -3) and (-2.4, -1.8)
Practice Assignment
• Block Page 218, 10 - 28 Even