Lines and Distance
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, (x1, y1) and (x2, y2), 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 (x2 – x1) and (y2 – y1).
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 double-check 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 (x1, y1) and (x2, y2), then the midpoint of the line segment is:
In other words, the x- and y-coordinates of the midpoint are the averages of the x- and y-coordinates of the endpoints.
Here’s a practice question:
What is the midpoint of the line segment whose endpoints are (6, 0) and (3, 7)?
To solve, all you have to do is plug the points given into the midpoint formula, x1 = 6, y1 = 0, x2 = 3, and y2 = 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 (x1, y1) and (x2, y2), 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 ma > mb > mc > md.
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,” y2y1 = 0, and thus, m = (y2y1)/(x2x1) = 0/(x2x1) = 0.
  • A line that is vertical has an undefined slope. In this case, there is no “run,” and x2x1 = 0. Thus, m = (y2y1)/(x2x1) = (y2y1)/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 y2y1 = x2x1 or y2 – y1= –(x2x1).
Of the four lines pictured below, one has a slope of 0, one has a slope of 1, another has a slope of –1, and another has undefined slope. Decide which is which.
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 slope-intercept form and the point-slope form.
Slope-Intercept Form
The slope-intercept form of the equation of a line is:
where m is the slope of the line and b is the y-intercept of the line. Both are constants.
The y-intercept of a line is the y-coordinate of the point at which the line intersects the y-axis. Likewise, the x-intercept of a line is the x-coordinate of the point at which the line intersects the x-axis. Therefore, if given the slope-intercept form of the equation of a line, you can find both intercepts.
For example, in order to find the y-intercept, simply set x = 0 and solve for the value of y. For the x-intercept, you set y = 0 and solve for x.
To sketch a line given in slope-intercept form, first plot the y-intercept, 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 y-intercept is at 3, and so the line crosses the y-axis at (0, 3). For practice, solve for the x-intercept. First, set y = 0, then:
Point-Slope Form
The point-slope form of the equation of a line is:
where m is the slope of the line and (x1, y1) is a point on the line.
The point-slope form and slope-intercept form are just alternative ways of expressing the same equation. In fact, the slope-intercept form is the point-slope form taken at the y-intercept, or the point (0, y1):
Since y1 = b (the y-intercept is simply the y-coordinate of the point at which x = 0), the two forms are equal.
The slope-intercept form of the line equation is the more common of the two, but the point-slope form is extremely useful when all the information you have is the slope and a point (hence “point-slope”).
Example Problems with Slope
The Math IIC test often asks questions that require you to understand the slope-intercept form and the point-slope form and to be able to convert between the two.
Here’s a practice question.
What is the slope-intercept equation of the line that contains the point (3,4) and is perpendicular to the line y = x – 6?
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 point-slope equation is y – 4 = –3(x – 3). To convert this to slope-intercept form, use algebra:
Here’s another question:
What is the slope-intercept form of the equation of the line that contains the points (5, 3) and (–1, 8)?
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 slope-intercept form, the only additional information we need is the y-intercept. To find it, plug in the x- and y-coordinates 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 slope-intercept form of the equation of this line is y = –5 /6 x + 43/6.
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