Introduction to Slope of a Line
A line is a dot that went for a walk.
—Paul Klee (1879–1940)
Now that you are familiar with the coordinate plane, this lesson takes you a little deeper into the world of coordinate geometry. You will discover slopes, midpoints, and the distance formula.
You now know how to plot points on a coordinate plane. When two points on the coordinate plane are connected, a line is formed. The slope of a line is the difference between the y values of two points divided by the difference between the x values of those two points.
The slope between two points (x_{1}, y_{1}) and (x_{2}, y_{2}) can be found by using the following formula:
Slope is known as "the rise over the run." This means that the number in the numerator (top number) tells how many units to move up or down and the number in the denominator (bottom number) tells how many units to move across to the right or left.
Use the formula for slope:
Here, the slope, or the rise over the run, is . As you learned in Lesson 4, a negative fraction, no matter where the negative sign is placed, makes the entire fraction negative. Therefore, you could also write the slope as .
Tip:If both the y value and the x value increase from one point to another, or if both the y value and the x value decrease from one point to another, the slope of the line is positive. If the y value increases and the x value decreases from one point to another, or if the y value decreases and the x value increases from one point to another, the slope of the line is negative. |
A horizontal line has a slope of 0. Lines such as y = 9 or y = –5 are lines with slopes of 0.
A vertical line has no slope. Lines such as x = 2 or x= –8 are lines with no slopes.
Midpoint
Any two points determine a segment on the coordinate plane. Every segment has a midpoint. The midpoint is exactly halfway between the two points.
Midpoint Formula
The coordinates of the midpoint of a segment, given the coordinates of its endpoints as (x_{1}, y_{1}) and (x_{2}, y_{2}), is midpoint = . This is called the midpoint formula.
Let's try a midpoint problem.
What is the midpoint of the segment with endpoints A (–3,–5) and B (–6,7)?
Plug the numbers into the midpoint formula.
M
= (–4.5,1)
You can also use the midpoint formula to find a missing coordinate.
What is the endpoint, C, of a segment whose midpoint, M, is (7,0) and other endpoint is D (10,4)?
Use the midpoint formula to find (x_{2}, y_{2}):
Multiply both sides by 2:
10 + x_{2} = 14
4 + y_{2} = 0
Now, subtract from both sides to isolate the variable:
x_{2} = 4
y_{2} = –4
The other endpoint is (4,–4).
Distance
To find the distance between two points, use the following formula. The variable x_{1} represents the x-coordinate of the first point, x_{2} represents the x-coordinate of the second point, y_{1} represents the y-coordinate of the first point, and y_{2} represents the y-coordinate of the second point:
D = √(x_{2}– x_{1}) 2 + (y_{2}– y_{1}) 2
What is the distance between the points (–2,8) and (4,–2)? Substitute these values into the formula:
D = √(4 – (–2))^{2} + ((–2) – 8)^{2}
D = √ (4 + 2)^{2} + (– 2 – 8)^{2}
D = √(6)^{2} + (– 10)^{2}
D = √36 + 100
D = √136
D = 2√34
Find practice problems and solutions for these concepts at Slope of a Line: Practice Questions.
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