162 Part 3: The Shape of the World
Slopes
Geometry and algebra may feel like different worlds at times, but now and then, they come
together, and that often happens on the coordinate plane. When we looked at graphing linear
equations, you saw how the slope of a line controlled its tilt or angle. When you talk about
parallel and perpendicular lines, the angles the lines make are important.
The graphs of two lines in the coordinate plane will be parallel lines if they have the same slope.
The matching slopes mean they run at the same angle and don’t tilt toward each other, so they
never cross. The line y = 2x – 3 and the line y = 2x + 5 both have a slope of 2, and so they will
be parallel.In order for the graphs of two linear equations to be perpendicular lines, one must rise and
one must fall, so the slopes must have opposite signs. That alone won’t get that exact right
angle, however. To actually be perpendicular, the lines must have slopes that are negative
reciprocals. If one line has a slope of 2, a line perpendicular to it will have a slope of -1
2. The
graphs of yx^3
5
5 - 14 and yx^5
3 1 are perpendicular lines because their slopes,^3
5- and^5
3
,
multiply to -1, so they are negative reciprocals.-6 -2 062
-2246-4-6-8-9 -7 -5 -4 -1-3 134 5 78 9-8
-9-1-3-5-7891357-10 10-1010yx