Slopes and collinearity connect algebra with the visual structure of a coordinate plane. A slope tells how steep a line is by comparing vertical change to horizontal change. Three or more points are collinear if they lie on the same straight line.
This idea matters because it lets you prove a geometric relationship using coordinates and arithmetic.
Key Facts
- Slope formula: m = (y2 - y1) / (x2 - x1).
- Three points A, B, and C are collinear if slope AB = slope BC = slope AC, when the slopes are defined.
- A vertical line has undefined slope because x2 - x1 = 0.
- If three points all have the same x-coordinate, they are collinear on a vertical line.
- Rise = change in y = y2 - y1, and run = change in x = x2 - x1.
- Equal slopes mean equal steepness and direction, so the points lie on the same straight path if they share a connecting line.
Vocabulary
- Slope
- Slope is the ratio of vertical change to horizontal change between two points on a line.
- Collinear
- Collinear points are points that lie on the same straight line.
- Coordinate plane
- A coordinate plane is a grid formed by a horizontal x-axis and a vertical y-axis.
- Rise
- Rise is the change in y-values between two points.
- Run
- Run is the change in x-values between two points.
Common Mistakes to Avoid
- Subtracting coordinates in different orders is wrong because the x-values and y-values must be subtracted in the same point order, such as (y2 - y1) / (x2 - x1).
- Comparing only two of the three slopes is incomplete because one equal pair of slopes does not always prove all three points are on one line unless the shared point and line relationship are checked correctly.
- Treating vertical slope as 0 is wrong because a vertical line has run 0, so its slope is undefined, not zero.
- Using decimals too early can hide equality because fractions such as 2/3 and 4/6 are exactly equal even if rounded decimals appear slightly different.
Practice Questions
- 1 Find the slopes AB, BC, and AC for A(1, 2), B(3, 6), and C(5, 10). Are the three points collinear?
- 2 Determine whether P(-2, 5), Q(1, -1), and R(4, -7) are collinear by comparing slopes.
- 3 A student says that points with slopes AB = 3 and AC = 3 must be collinear. Explain why using the same starting point A helps support the conclusion, and describe what would be different if the equal slopes came from unrelated pairs of points.