Determining Collinearity of Points on an XY Plane
To determine whether three given points are collinear (lie on the same straight line), one can use several mathematical methods. In this article, we will explore a detailed explanation of how to show that the points 44, 02, and -2-5 are not collinear using the slope method, line equation, and the shoelace formula.
The Slope Method
For points A, B, and C to be collinear, the slope of the line segments AB, AC, and BC must all be equal. If any of these slopes are different, the points are not collinear.
Example: Points 44, 02, and -2-5
Let us take the points A(4, 4), B(0, 2), and C(-2, -5) and determine if they are collinear.
Slope of AB
[text{Slope of AB} frac{2 - 4}{0 - 4} frac{-2}{-4} frac{1}{2}]Slope of AC
[text{Slope of AC} frac{-5 - 4}{-2 - 4} frac{-9}{-6} frac{3}{2}]Slope of BC
[text{Slope of BC} frac{-5 - 2}{-2 - 0} frac{-7}{-2} frac{7}{2}]Since the slopes are not equal ((frac{1}{2} eq frac{3}{2} eq frac{7}{2})), we can conclude that points 44, 02, and -2-5 are not collinear.
Line Equation Method
A more formal approach involves finding the equation of the line that passes through two points and then checking if the third point satisfies this equation.
Example: Finding the Equation of the Line AB
The slope-intercept form of a line equation is given by y mx b.
Using points A(4, 4) and B(0, 2), we can determine the slope m and the y-intercept b.
Slope (m)
[m frac{2 - 4}{0 - 4} -frac{1}{2}]The equation of the line is then:
[y -frac{1}{2}x 2]Checking Point C(-2, -5)
[text{Substitute } x -2 text{ into } y -frac{1}{2}x 2 Rightarrow y -frac{1}{2}(-2) 2 3]Since point (-2, -5) does not satisfy the equation y 3, the points are not collinear.
Shoelace Formula
The shoelace formula, also known as Gauss's area formula, can be used to determine the area of a polygon given its vertices. If the points form a degenerate triangle (a straight line), the area will be zero. Here, we calculate the area to determine collinearity.
Calculating Area Using the Shoelace Formula
The area (A) of a polygon with vertices (x1, y1), (x2, y2), ..., (xn, yn) is given by:
[A frac{1}{2}left| (x_1y_2 - x_2y_1) (x_2y_3 - x_3y_2) ... (x_{n-1}y_n - x_ny_{n-1}) (x_ny_1 - x_1y_n) right|]For points 44, 02, and -2-5, we calculate:
[A frac{1}{2}| (4 times 2 0 times -5 - 2 times 4) - (4 times 0 2 times -2 -5 times 4)| frac{1}{2}| 8 - 8 - (0 - 4 - 20)| frac{1}{2}| 0 - 24 | frac{1}{2} times 24 12 text{ square units}]Since the area is not zero, the points are not collinear.
Conclusion
In conclusion, using the slope method, line equation method, and the shoelace formula, we have shown that points 44, 02, and -2-5 are not collinear. This confirms that these points are not aligned on a straight line.
Keywords: collinearity, slope, shoelace formula, line equation