The points A(x, 2), B(2, 1) and C(6, 3) are collinear in the coordinate plane. What is the value of x?

Introduction / Context:
This coordinate geometry question checks your understanding of collinearity of points. Three points are collinear if they lie on a single straight line. One standard method to test collinearity is to compare slopes of any two segments formed by the points.

Given Data / Assumptions:

• Points are A(x, 2), B(2, 1), and C(6, 3).
• All three points are given to be collinear.
• We must determine the unknown x coordinate of point A.

Concept / Approach:
For points A, B, and C to be collinear, the slope of AB must equal the slope of BC. The slope between two points (x1, y1) and (x2, y2) is: m = (y2 - y1) / (x2 - x1) We compute the slope of BC, then set the slope of AB equal to this value, and solve for x.

Step-by-Step Solution:
Compute slope of BC: B(2, 1), C(6, 3) m_BC = (3 - 1) / (6 - 2) = 2 / 4 = 1/2 Compute slope of AB: A(x, 2), B(2, 1) m_AB = (1 - 2) / (2 - x) = -1 / (2 - x) Set slopes equal for collinearity: -1 / (2 - x) = 1/2 Cross multiply: -2 = 2 - x Rearrange: -2 - 2 = -x, so -4 = -x Hence x = 4

Verification / Alternative check:
With x = 4, point A becomes (4, 2). Compute slope from B(2, 1) to A(4, 2): (2 - 1) / (4 - 2) = 1/2, which matches slope of BC. Alternatively, we can check that the three points satisfy a linear equation y = (1/2)x. For x = 2 and x = 6, we get y = 1 and y = 3, matching B and C. For x = 4, y = 2, matching the computed point A.

Why Other Options Are Wrong:
If x = -2, then A(-2, 2) leads to slope AB = (1 - 2) / (2 - (-2)) = -1/4, which does not equal 1/2. For x = -4 or x = 0 or x = 2, similar calculations give slopes different from 1/2, so those values do not make the three points collinear.

Common Pitfalls:
Typical mistakes include reversing the order of subtraction when computing slopes, leading to sign errors, or equating slopes of wrong pairs of points. Some students also incorrectly cross multiply. Writing each slope carefully and verifying with a simple final substitution helps avoid errors.

Final Answer:
The value of x that makes the three points collinear is 4.