Collinear Points
Collinear points are three or more points that lie exactly on the same straight line. In coordinate geometry, these points do not enclose any space, meaning they cannot form a triangle.
Key Concepts
For three points A, B, and C to be collinear, the sum of the distances between two pairs of points must equal the distance between the extreme points, such as AB + BC = AC.
Because collinear points lie on a single flat path, they cannot serve as the vertices of a two-dimensional shape like a polygon.
When applying the coordinate geometry area formula, the area of a triangle formed by any three collinear points will always calculate to exactly zero.
Formula & Equation
Area = 0.5 * [x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)] = 0 OR AB + BC = AC
(x1, y1), (x2, y2), and (x3, y3) are the coordinates of the three points. AB, BC, and AC represent the distances between these points.
Common Misconceptions
Myth: Any three points plotted on a graph can be connected by a line, so they are always collinear.
Fact: While any two points are always collinear because a single straight line can connect them, three points are only collinear if they lie on the exact same straight line. If they do not, connecting them will form a triangle.
Real World Applications
Surveyors placing three boundary markers in a perfectly straight line to define a property edge.
Astronomers observing a syzygy, an event where the Sun, Earth, and Moon align in a straight line during an eclipse.
Frequently Asked Questions
How do you prove three points are collinear using the distance formula?
Calculate the distances between all three pairs of points. If the sum of the two smaller distances equals the largest distance, the points are collinear.
What is the area of a triangle formed by collinear points?
The area is always zero because the points lie on a flat line and cannot enclose any space to form a triangle.