All you need to understand about straight lines is contained in
the picture above. Think of moving along the blue line. You climb
one unit for every two units that you move to the right. The slope of
the line is the ratio of the vertical and the horizontal distances,
called **rise** and **run**, respectively.

A straight line is characterized by the fact that the slope is
independent of where you compute the rise and the run. The two
triangles ** AEC** and ** BFD** in the Figure are ** similar.**
This means that corresponding
angles are equal, and hence the ratios of the lengths of
corresponding sides are equal. It does not matter which two points on
the line we pick (as long as they are distinct), we will get a similar
triangle, and the same slope, for all choices. In the picture,
the various points have the following coordinates:

Using the cyan triangle ** AEC** we obtain for the slope of the line

**Everything about straight lines flows from the concept of slope.**
There are some details and some language that you need to be familiar
with. But essentially all problems concerning straight lines,
certainly in this class, require that you work with the slope of the
straight line.

A straight line may be horizontal in which case its slope is zero. It may be vertical, in which case the definition of slope breaks down since it calls for a division by zero. We say that the slope is undefined. It may also descend to the right in which case the slope is negative. Check here for an example of a line with a negative slope.

A straight line may intersect the and axes, in the
points and , respectively. The numbers
and are called the -**intercept** and -**intercept** of
the straight line, respectively. Note that the intercepts are
** numbers**, they are ** not points**. For the line illustrated on this page
the intercept =
and the -intercept is
.

The line in the above Figure is described by the equation

Suppose we are given two points and on a line. Let's denote a general point on the line by . Then, since the slope is independent of the choice of points we obtain the equation

If we are given the slope of a line, and a point on it, we denote again by a general point and obtain

An equation of the form

Two lines are parallel if their slopes are equal. You can see this immediately by drawing two parallel lines and triangles like those in the Figure on this page. All the triangles are similar.

Two lines with slopes and are perpendicular if

The following Figure explains why the slopes are related in this way:

It shows two perpendicular lines with slopes **1/2** (shown in
blue) and
**-2** (shown in red), intersecting in the point **(2,3)**. The
two green triangles are congruent, and the one associated with the red
line can be obtained by rotating by 90 degrees the other triangle
associated with the blue line. If you think of subtracting the
coordinates of the point close to the intersection from those of the
point far away, that rotation turns the original run into the new
rise, and the original rise into the negative of the new run, as
illustrated in the Figure. You can also think of subtracting the
coordinates of the leftmost point from those of the rightmost, in
which case the rise is multiplied with negative 1 and the run
maintains its sign, without affecting the sign of the slope.