Chapter 4-4 Parallel and Perpendicular Lines
Parallel Lines:
2 lines that are parallel to each other have the SAME SLOPE
Example:
y = 2x – 10
y = 2x + 4
Are parallel because they both have a slope of 2
BE CAREFUL:
to be parallel they must have the SAME SLOPE BUT DIFFERENCE y-INTERCEPTS
to be parallel they must have the SAME SLOPE BUT DIFFERENCE y-INTERCEPTS
If they have the same m and the same b then they are COLLINEAR ( or they COINCIDE) they are the same line and NOT parallel
Perpendicular Lines
2 lines that are perpendicular to each other have SLOPES that are:
Opposite signs
and
Reciprocals
and
Reciprocals
y = 2x – 10 is perpendicular to y = (-1/2)x + 4
This is another twist to our Mystery Line Puzzles…
If you know that the mystery line is parallel or perpendicular to another given line they you know the mystery line’s slope!!
EXAMPLE 1:
Your mystery line has a point of (2, -5) and is PARALLEL to the line y = 2x + 4
So you know that the mystery line’s slope
because it is the same as the given line (m = 2)
because it is the same as the given line (m = 2)
Substitute the slope and the point given for the mystery line and solve for b OR use Point Slope Form and solve easily!
This is why I like Point Slope: y + 5 = 2(x – 2)
This is why I like Point Slope: y + 5 = 2(x – 2)
y + 5 = 2x – 4
y = 2x – 9
If I used Slope Intercept form or y = mx + b
-5 = 2(2) + b
-5 = 4 + b
b = -9
Plug everything back in
y = 2x – 9
Same results—but I think that it takes longer (students also tend to FORGET to put everything together)
EXAMPLE 2:
Your mystery line has a point of (2, -5) and is perpendicular to the line y = 2x + 4
So you know that the mystery line’s slope is the opposite sign and the reciprocal of the given line
(since the given line’s slope is 2, the mystery line’s slope must be –½ )
(since the given line’s slope is 2, the mystery line’s slope must be –½ )
Substitute the slope and the point given on the mystery line and solve for b OR use Point Slope Form and solve easily!
y + 5 = (-½)(x -2)
y + 5 = -½x + 1
y = -½x - 4
If I used Slope Intercept form or y = mx + b
-5 = (-½)(2) + b
-5 = -1 + b
-4 = b
Plug everything back
y = -½x - 4
