3-6
LESSON 3 –6
Perpendiculars and
Distance
Five-Minute Check (over Lesson 3
Key Concept: Distance Between a Point and a Line
Postulate 3.6: Perpendicular Postulate
Example 1: Real-World Example: Construct Distance From
Example 2: Distance from a Point to a Line on Coordinate Plane
Key Concept: Distance Between Parallel Lines
Theorem 3.9: Two Line Equidistant from a Third
Example 3: Distance Between Parallel Lines
Over Lesson 3 –5
Given
9
13, which segments are parallel?
___ ___
A.
AB || CD
___ __
B.
FG || HI
___ ___
C.
CD || FG
D.
none
Over Lesson 3 –5
Given
2
5, which segments are parallel?
___ ___
A.
AB || CD
___ ___
B.
CD || FG
___ __
C.
FG || HI
D.
none
Over Lesson 3 –5
___ ___
If m
2 + m
4 = 180, then AB || CD. What postulate supports this?
A.
If consecutive interior
s are supplementary, lines are ||.
B.
If alternate interior
s are
, lines are ||.
C.
If corresponding
s are
, lines are ||.
D.
If 2 lines cut by a transversal so that corresponding
s are
, then the lines are ||.
Over Lesson 3 –5
___ __
If
5
14, then CD || HI. What postulate supports this?
A.
If corresponding
s are
, lines are ||.
B.
If 2 lines are to the same line, they are ||.
C.
If alternate interior
s are
, lines are ||.
D.
If consecutive interior
s are supplementary, lines are ||.
Over Lesson 3 –5
___
.
1 = 4x + 6 and m
14 = 7x – 27.
__
A.
6.27
B.
11
C.
14.45
D.
18
Over Lesson 3 –5
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
Targeted TEKS
G.2(B) Derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines.
Mathematical Processes
G.1(E), G.1(F)
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.
• equidistant
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 .
Construct Distance From Point to a Line
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?
A.
AD
B.
AB
C.
CX
D.
AX
Distance from a Point to a Line on Coordinate Plane
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 .
Begin by finding the slope of the line through points (0, 0) and
( –5, 5).
( –5, 5)
V (1, 5)
(0, 0)
Distance from a Point to a Line on Coordinate Plane
Then write the equation of this line by using the point
(0, 0) on the line.
Slope-intercept form m = –1, ( x
1
, y
1
) = (0, 0)
Simplify.
The equation of line s is y = –x .
Distance from a Point to a Line on Coordinate Plane
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, ( x
1
, y
1
) = (1, 5)
Simplify.
Subtract 1 from each side.
The equation of line t is y = x + 4.
Distance from a Point to a Line on Coordinate Plane
Step 3 Solve the system of equations to determine the point of intersection.
line s : y = – x line t : (+) y = x + 4
2 y = 4 Add the two equations.
y = 2 Divide each side by 2.
Solve for x .
2 = – x Substitute 2 for y in the first equation.
–2 = x Divide each side by –1.
The point of intersection is ( –2, 2). Let this point be Z .
Distance from a Point to a Line on Coordinate Plane
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 or about 4.24 units.
A.
B.
C.
D.
COORDINATE GEOMETRY
Line n contains points (2, 4) and ( –4, –2). Find the distance between line n and point B(3, 1).
( –4, 2)
(2, 4)
B ( 3 , 1 )
Distance Between Parallel Lines
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 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.
a b p
Sketch line p through the yintercept of line b , (0, –1), perpendicular to lines a and b .
Distance Between Parallel Lines
Step 1
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.
Distance Between Parallel Lines
Step 2
Use a system of equations to determine the point of intersection of the lines a and p .
Substitute 2 x + 3 for y in the second equation.
Group like terms on each side.
Distance Between Parallel Lines
Simplify on each side.
Multiply each side by .
Substitute equation for p.
for x in the
Distance Between Parallel Lines
Simplify.
The point of intersection is or ( –1.6, –0.2).
Distance Between Parallel Lines
Step 3
Use the Distance Formula to determine the distance between (0, –1) and (–1.6, –0.2).
Distance Formula x
2 y
2
= –1.6, x
1
= –0.2, y
1
= 0,
= –1
Answer: The distance between the lines is about
1.79 units.
Find the distance between the parallel lines a and b whose equations are and , respectively.
A.
2.13 units
B.
3.16 units
C.
2.85 units
D.
3 units
