Condition of Collinearity of 3 Points

Collinearity refers to the condition where three or more points lie on the same straight line. In the context of coordinate geometry, we can determine whether three points are collinear by using various methods. Understanding the condition of collinearity is crucial for solving problems related to straight lines, geometric proofs, and coordinate geometry.

Mathematical Representation

Given three points (A(x_1, y_1)), (B(x_2, y_2)), and (C(x_3, y_3)), these points are collinear if and only if the area of the triangle formed by these points is zero. This is because if the points are on the same line, the "triangle" they form will have no area.

The area of a triangle given three vertices can be calculated using the determinant formula:

[ \text{Area} = \frac{1}{2} \left| \begin{array}{ccc} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \ \end{array} \right| ]

For collinearity, this determinant must be equal to zero:

[ \left| \begin{array}{ccc} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \ \end{array} \right| = 0 ]

Slope Method

Another method to check for collinearity is to compare the slopes of the line segments joining the points. If the slopes of (AB), (BC), and (AC) are equal, then the points are collinear.

The slope of the line through points (P(x_1, y_1)) and (Q(x_2, y_2)) is given by:

[ m_{PQ} = \frac{y_2 - y_1}{x_2 - x_1} ]

For points (A), (B), and (C) to be collinear, the following condition must be satisfied:

[ m_{AB} = m_{BC} = m_{AC} ]

Differences and Important Points

Aspect	Area Method	Slope Method
Basis	Based on the area of the triangle formed by the points	Based on the equality of slopes between pairs of points
Calculation	Involves calculating a determinant	Involves calculating the slope for each pair of points
Applicability	Can be used even if two points have the same x-coordinate	Cannot be used directly if the line is vertical (slope is undefined)
Formula	(\frac{1}{2} \left	\begin{array}{ccc} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \ \end{array} \right
Geometric Interpretation	Relies on the concept of area	Relies on the concept of slope (rate of change)

Examples

Example 1: Using the Area Method

Determine if the points (A(1, 2)), (B(3, 6)), and (C(5, 10)) are collinear.

[ \text{Area} = \frac{1}{2} \left| \begin{array}{ccc} 1 & 2 & 1 \ 3 & 6 & 1 \ 5 & 10 & 1 \ \end{array} \right| = \frac{1}{2} \left( (1 \cdot 6 \cdot 1 + 2 \cdot 1 \cdot 5 + 1 \cdot 3 \cdot 10) - (1 \cdot 3 \cdot 10 + 2 \cdot 5 \cdot 1 + 1 \cdot 6 \cdot 1) \right) = 0 ]

Since the area is zero, the points are collinear.

Example 2: Using the Slope Method

Determine if the points (A(1, 2)), (B(3, 6)), and (C(5, 10)) are collinear.

[ m_{AB} = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2 ] [ m_{BC} = \frac{10 - 6}{5 - 3} = \frac{4}{2} = 2 ] [ m_{AC} = \frac{10 - 2}{5 - 1} = \frac{8}{4} = 2 ]

Since (m_{AB} = m_{BC} = m_{AC}), the points are collinear.

In both examples, we have shown that the points (A(1, 2)), (B(3, 6)), and (C(5, 10)) are collinear using two different methods. The consistency of the results across both methods confirms the collinearity of the points.