Determine x for Collinear Points on a Linear Path
Understanding Collinear Points and Their Importance in Geometry
In geometry, points are considered collinear if they lie on the same straight line. The determination of collinearity can be approached through various mathematical tools, one of which is slope calculations. This article explores the method to determine the value of x that makes the points (1, 2), (-1, x), and (2, 3) collinear. We'll delve into the step-by-step process, using the concept of slope and linear equations.
Conceptualizing the Problem
The problem at hand requires us to find the value of x for which the points (1, 2), (-1, x), and (2, 3) are collinear. Collinear points share a common slope, which is the ratio of the change in vertical distance (rise) to the change in horizontal distance (run).
Setting Up the Equations
To solve the problem, we can represent the given points as follows:
A (1, 2) B (-1, x) C (2, 3)We can use the slope formula to equate the slopes between the points. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
m (y2 - y1) / (x2 - x1)
Calculating the Slope of AB
The slope of AB (between points A and B) can be calculated as:
mAB (x - 2) / ((-1) - 1) (x - 2) / -2
Calculating the Slope of BC
The slope of BC (between points B and C) can be calculated as:
mBC (3 - x) / (2 - (-1)) (3 - x) / 3
Since AB and BC are collinear, their slopes are equal:
mAB mBC
Hence, we set the two slopes equal to each other:
(x - 2) / -2 (3 - x) / 3
Solving the Equation
To solve for x, we can cross-multiply to eliminate the denominators:
3(x - 2) -2(3 - x)
Distributing the constants on both sides of the equation:
3x - 6 -6 2x
Next, we isolate x by adding 6 to both sides and subtracting 2x from both sides:
3x - 2x -6 6
x 3
Therefore, for the points (1, 2), (-1, x), and (2, 3) to be collinear, the value of x must be 3.
Conclusion
The determination of the value of x for collinear points involves the use of linear equations to ensure that the slopes between the given points are equal. The problem was solved systematically by representing the points and calculating the slopes, leading to the conclusion that x equals 3. Understanding these concepts can greatly enhance your problem-solving skills in geometry and linear algebra.