Transcript Ch 3-6 Perpendiculars and Distance
Advanced Topic for 9 th & 10 th only
Chapter 3-6 Perpendiculars and Distance
Distance between a Point and a Line:
The distance between a point and a line, is the length of the segment perpendicular to the line from the point.
C
Shortest distance
B A
Which segment in the diagram
represents the distance from R to
XY?
A.
___ RY ___
B.
___
C.
D.
___
Equidistant: same distance.
a
Theorem:
In a plane if two lines are equidistant from a third line, then the two lines are parallel to each other. If the distance d between line a and b is d and distance between b d b and c is d then a and c are Parallel.
c
Find the distance between the parallel lines
y
1
x
3 3 and
y
1
x
3 1 3
Graph the original two equations.
y
3 1
x
3 and
y
1 3
x
1 3 -9 -8 -7 -6 -5 -4 -3 -2 -1 9 8 7 6 5 4 3 2 1 -1 1 -2 -3 -4 -5 -6 -7 -8 -9 2 3 4 5 6 7 8 9
y
y
m
(
x
x
) the line perpendicular to the original two equations.
Use one of the y intercepts of the original equations.
y y
y
1 ( 3 )
m
(
x
3 (
x x
1 ) 0 )
y
3
x
3 9 8 7 6 5 4 3 2 1 So the equation of the green line is
y
3
x
3 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 -2 -3 -4 -5 -6 -7 -8 -9 2 3 4 5 6 7 8 9
Use system of equations to determine where the green line intersects the top blue equation.
y y
3 1 3
x x
3 1 3 9 8 7 6 5 1 3
x
1 3 = 3
x
3 4 3 2 1 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 3
x
3
x
3 1 3 10 3
x x
10 3 1 -1 1 -2 -3 -4 -5 -6 -7 -8 -9 2 3 4 5 6 7 8 9
Now you know that at x=1 the green graph crosses the graph on top, plug in x=1 into the equation of the green line.
y
3
x
3
y
3 ( 1 ) 3
y
0
The intersection point is (1,0)
Now use the distance formula:
d
Between points (0,-3) and (1,0).
(
x
2
x
1 ) 2 (
y
2
y
1 ) 2
d
(
x
2
x
1 ) 2 (
y
2
y
1 ) 2
d
( 0 1 ) 2 ( 3 0 ) 2
d
10 -9 -8 -7 -6 -5 -4 -3 -2 -1 9 8 7 6 5 4 3 2 1 -1 1 -2 -3 -4 -5 -6 -7 -8 -9 2 3 4 5 6 7 8 9
Homework
• Textbook pages 185 – 187, • problems 1, 4 – 7, 10 – 18 evens, • and 36 – 42 evens.