# 3-6 Perpendiculars and Distance

```Ms. Andrejko
Real World
Vocabulary
 Equidistant- the distance between two lines
measured along a perpendicular line to the lines is
always the same
 The distance between a line and a point not on the line
is the length of the segment perpendicular to the line
from the point.
 The distance between two parallel lines is the
perpendicular distance between one of the lines and
any point on the other line.
Theorems &amp; Postulates
 Postulates
 3.6
 Theorem
 3.9 – Two Lines Equidistant from a Third- in a plane,
if two lines are equidistant from a third line, then the
two lines are parallel to each other.
Examples
 Construct the segment that represents the distance
indicated.
Examples
 Construct the segment that represents the distance
indicated.
Steps (when given line and pt.)
 1. Find perpendicular slope
 2. Make equation using perpendicular slope and given
point (P)
 3. Solve system of equations to find point of
intersection
 4. Plug into distance formula
Examples
 Find the distance from P to ℓ.
1.
Line ℓ y=-x-2. Point P has coordinates (2, 4)
Perpendicular slope:
m=1 Equation through point:
y-4 = 1(x-2)
y-4 = x-2
y=x+2
-x-2=x+2 y=x+2
System of equations: -2=2x+2 y=-2+2
y=0
-4=2x
(-2,0)
-2=x
Distance formula:
√(-2-2)2+(0-4)2
√32 = 4√2
(x 2  x1)2  (y 2  y1)2
√(-4)2+(-4)2
√16+16
Practice
 Find the distance from P to ℓ.
1.
Line ℓ y=-x. Point P has coordinates (1, 5)
Perpendicular slope:
m=1 Equation through point:
-x = x+4
System of equations: -2x=4
x=-2
y=-2+4
y=2
(-2,2)
y-5 = 1(x-1)
y-5 = x-1
y= x+4
Distance formula:
(x 2  x1)2  (y 2  y1)2
√(-2-1)2+(2-5)2
√(-3)2+(-3)2
√9+9
√18 = 3√2
Example
 Find the distance from P to ℓ.
1.
Line ℓ y=(4/3)x-2. Point P has coordinates (-1, 5)
Perpendicular slope:
System of equations:
Distance formula:
(x 2  x1)2  (y 2  y1)2
Equation through point:
Practice
 Find the distance from P to ℓ.
1.
Line ℓ y=-3x+8. Point P has coordinates (-1,1)
Perpendicular slope:
System of equations:
Distance formula:
(x 2  x1)2  (y 2  y1)2
Equation through point:
Steps (when given 2 parallel equations)
 1. Find y-intercept of line m
 2. Write equation of perpendicular line through y-
intercept (line p).
 3. Use systems of equations to determine point of
intersection between l and p.
 4. Use distance formula
Examples
 Find the distance between each pair of parallel lines
with the given equations.
y=5x-22
y=5x+4
Practice
 Find the distance between each pair of parallel lines
with the given equations.
y=-3x+3
y=-3x-17
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