PARALLEL AND PERPENDICULAR LINES

This web exercise explores the connection between slope and parallel/perpendicular lines.

Two lines in a plane are parallel if they never intersect; they have the same ‘slant’.
(For a more rigorous study of parallel lines and related concepts, visit my Geometry course: Parallel Lines.)
Slope (or lack of it!) can be used to decide if lines are parallel:

CHARACTERIZATION OF PARALLEL LINES INVOLVING SLOPE
Suppose two lines lie in the same coordinate plane.
Then,
the two lines are parallel
if and only if
(they are both vertical)     or     (they have the same slope)

This statement offers a good opportunity to review the mathematical words ‘if and only if’ and ‘or’.
The statement has been visually presented to help you see the mathematical sentence structure more clearly.

Now, let's move on to perpendicular lines.
Two lines are perpendicular if they intersect at a $\,90^{\circ}$ angle.
For example, the $\,x$-axis and $\,y$-axis are perpendicular.

CHARACTERIZATION OF PERPENDICULAR LINES
INVOLVING SLOPE
Suppose two non-vertical lines lie in the same coordinate plane.
Let $\,m_1\,$ and $\,m_2\,$ denote the slopes of these lines.

The two lines are perpendicular if and only if their slopes are opposite reciprocals: $$\,m_1 = -\frac{1}{m_2}\,$$
Equivalently, the two lines are perpendicular if and only if their slopes multiply to $\,-1\,$: $$\,m_1m_2 = -1\,$$

Make sure you understand the opposite reciprocal equation:

$$ m_1 = -\frac{1}{m_2} $$
one of the slopes is the opposite of the reciprocal of the other slope
$\displaystyle m_1$ $\displaystyle =$ $\displaystyle -$ $\displaystyle\frac{1}{m_2}$

For example, what's the reciprocal of $\,2\,$? Answer:   $\frac12$
What's the opposite [of the] reciprocal of $\,2\,$? Answer:   $-\frac12$
So, the numbers $\,2\,$ and $\,-\frac 12\,$ are opposite reciprocals.
Also notice that $\,(2)(-\frac 12) = -1\,$.
So, lines with slopes $\,2\,$ and $\,-\frac12\,$ are perpendicular.

It's easy to see that this is the correct characterization for perpendicular lines, by studying the sketch below:

The yellow triangle, with base of length $\,1\,$ and right side of length $\,m\,$,
shows that the slope of the first line is $\,\frac{\text{rise}}{\text{run}} = \frac{m}1 = m\,$.

Now, imagine that this yellow triangle is a block of wood that is glued to the line.
Rotate this block of wood counter-clockwise by $\,90^{\circ}\,$ (so that the original base is now vertical).
Using the rotated triangle to compute the slope of the new, rotated, line gives:   $\,\frac{\text{rise}}{\text{run}} = \frac{1}{-m} = -\frac1m\,$
Easy! Voila!

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Linear Inequalities in Two Variables
CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
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11 12 13 14 15 16 17 18 19 20
AVAILABLE MASTERED IN PROGRESS

(MAX is 20; there are 20 different problem types.)