


Lines and Distance
Lines and distance are fundamental to coordinate geometry,
not to mention to the Math IIC test. Even the most complicated coordinate
geometry question will use the concepts covered in the next couple
ofsections.
Distance
Measuring distance in the coordinate plane is made possible
thanks to the Pythagorean theorem. If you are given two points,
(x_{1}, y_{1})
and (x_{2}, y_{2}),
their distance from each other is given by the following formula:
The diagram below shows how the Pythagorean theorem plays
a role in the formula. The distance between two points can be represented
by the hypotenuse of a right triangle whose legs are of lengths
(x_{2} – x_{1})
and (y_{2} – y_{1}).
To calculate the distance between (4, –3) and (–3, 8),
plug the coordinates into the formula:
The distance between the points is , which equals approximately 13.04.
You can doublecheck this answer by plugging it back into the Pythagorean
theorem.
Finding Midpoints
The midpoint between two points in the coordinate plane
can be calculated using a formula. If the endpoints of a line segment
are (x_{1}, y_{1})
and (x_{2}, y_{2}),
then the midpoint of the line segment is:
In other words, the x and ycoordinates
of the midpoint are the averages of the x and ycoordinates
of the endpoints.
Here’s a practice question:

To solve, all you have to do is plug the points given
into the midpoint formula, x_{1} = 6, y_{1} = 0, x_{2} = 3,
and y_{2} = 7:
Lines
Lines may be nothing more than an infinite set of points
arrayed in a straight formation, but there are a number of ways
to analyze them. We’ll look at some of the main properties, formulas,
and rules of lines.
Slope
The slope of a line is a measurement of how steeply the
line climbs or falls as it moves from left to right. More technically,
it is a line’s vertical change divided by its horizontal change, informally
known as “the rise over run.” Given two points on a line, call them
(x_{1}, y_{1})
and (x_{2}, y_{2}),
the slope of that line can be calculated using the following formula:
The variable most often used to represent slope is m.
So, for example, the slope of a line that contains the
points (–2, –4) and (6, 1) is:
Positive and Negative Slopes
You can easily determine whether the slope of a line is
positive or negative just by looking at the line. If a line slopes
uphill as you trace it from left to right, the slope is positive.
If a line slopes downhill as you trace it from left to right, the
slope is negative. You can get a sense of the magnitude of the slope
of a line by looking at the line’s steepness. The steeper the line,
the more extreme the slope will be; the flatter the line, the smaller
the slope will be. Note that an extremely positive slope is bigger than
a moderately positive slope while an extremely negative slope is smaller than
a moderately negative slope.
Look at the lines in the figure below and try to determine
whether the slope of each line is negative or positive and which
has a greater slope:
Lines a and b have positive
slopes, and lines c and d have
negative slopes. In terms of slope magnitude, line m_{a} > m_{b} > m_{c} > m_{d}.
Special Slopes
For the Math IIC, there are a few slopes you should recognize
by sight. If you can simply see one of these lines and identify
its slope without having to do any calculations you will save yourself
a lot of time.
 A line that is horizontal has a slope of zero. Since there is no “rise,” y_{2} – y_{1} = 0, and thus, m = (y_{2} – y_{1})/(x_{2} – x_{1}) = 0/(x_{2} – x_{1}) = 0.
 A line that is vertical has an undefined slope. In this case, there is no “run,” and x_{2}–x_{1} = 0. Thus, m = (y_{2} – y_{1})/(x_{2} – x_{1}) = (y_{2} – y_{1})/0 and any fraction with 0 in its denominator is, by definition, undefined.
 A line that makes a 45º angle with a horizontal has a slope of 1 or –1. This makes sense because the “rise” equals the “run,” and y_{2} – y_{1} = x_{2} – x_{1} or y2 – y_{1}= –(x_{2} – x_{1}).

Line a has slope 0 because it is horizontal.
Line b has slope –1 because it slopes downward at
45º as you move from left to right. Line c has
slope 1 because it slopes upward at 45ºas you move from left to
right. Line d has undefined slope because it is
vertical.
Parallel and Perpendicular Lines
Parallel lines are lines that don’t intersect. In coordinate
geometry, they can also be described as lines with the same slope.
Perpendicular lines are lines that intersect at a right
angle. In coordinate geometry, perpendicular lines have opposite,
reciprocal slopes. That is, a line with slope m is
perpendicular to a line with a slope of –(^{1}
/_{m} ).
In the figure, lines q and r both
have a slope of 2, so they are parallel. Line s is
perpendicular to both lines q and r and
thus has a slope of –(^{1}/
_{2} ).
Equation of a Line
A line in coordinate geometry can be described by an equation
containing the variables x and y.
For the Math IIC, you need to thoroughly understand two forms of
the equation of a line: the slopeintercept form and the pointslope
form.
SlopeIntercept Form
The slopeintercept form of the equation of a line is:
where m is the slope of the line and b is
the yintercept of the line. Both are constants.
The yintercept of a line is the ycoordinate
of the point at which the line intersects the yaxis.
Likewise, the xintercept of a line is the xcoordinate
of the point at which the line intersects the xaxis.
Therefore, if given the slopeintercept form of the equation of
a line, you can find both intercepts.
For example, in order to find the yintercept,
simply set x = 0 and solve for the value of y.
For the xintercept, you set y =
0 and solve for x.
To sketch a line given in slopeintercept form, first
plot the yintercept, and then use the slope of
the line to plot another point. Connect the two points to form your
line. In the figure below, the line y = –2x +
3 is graphed.
Since the slope is equal to –2, the line descends 2 units
for every 1 unit it moves in the positive x direction.
The yintercept is at 3, and so the line crosses
the yaxis at (0, 3). For practice, solve for the xintercept.
First, set y = 0, then:
PointSlope Form
The pointslope form of the equation of a line is:
where m is the slope of the line and
(x_{1}, y_{1})
is a point on the line.
The pointslope form and slopeintercept form are just
alternative ways of expressing the same equation. In fact, the slopeintercept
form is the pointslope form taken at the yintercept,
or the point (0, y_{1}):
Since y_{1} = b (the yintercept
is simply the ycoordinate of the point at which x =
0), the two forms are equal.
The slopeintercept form of the line equation is the more
common of the two, but the pointslope form is extremely useful
when all the information you have is the slope and a point (hence
“pointslope”).
Example Problems with Slope
The Math IIC test often asks questions that require you
to understand the slopeintercept form and the pointslope form
and to be able to convert between the two.
Here’s a practice question.

To answer this question, you need to first find the slope
of the line whose equation you are trying to write. Fortunately,
the question gives you the slope of a perpendicular line, and we
know that the slope of a line is the opposite reciprocal of the
slope of the line to which it is perpendicular. Thus, the slope
is ^{–1}⁄ = –3. If the line contains the point
(3, 4), its pointslope equation is y – 4 = –3(x –
3). To convert this to slopeintercept form, use algebra:
Here’s another question:

Start by finding the slope of the line. You can calculate
the slope with the two points you’re given: m =
^{8–3}/_{–1–5} =
–^{5}/_{6} .
To put the equation of this line in slopeintercept form, the only additional
information we need is the yintercept. To find
it, plug in the x and ycoordinates
of a point that you know is on the line into the equation y =
–^{5}/_{6 } x + b and
solve for b. Using the point (5, 3):
The slopeintercept form of the equation of this line
is y = –^{5}
/_{6 } x +
^{43}/_{6} .
