Step-by-Step Solution
Collinear means zero area: (−1)(4 − 12) + 2(12 − (−2)) + m((−2) − 4) = 0 → 8 + 28 − 6m = 0 → 6m = 36 → m = 6, so C = (6, 12). Check by steps: A to B moves 3 across and 6 up, B to C moves 4 across and 8 up — both are in the ratio 1 : 2, so the points line up. Assuming B is the MIDPOINT of AC gives m = 2(2) − (−1) = 5, but nothing in the question says that; a sign slip on ((−2) − 4) gives −6.
Answer: C) 6
Key idea: Area of a triangle from vertices
Given A(x₁, y₁), B(x₂, y₂), C(x₃, y₃), the area is:
**Area = 21∣x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|**
The absolute value sign |·| ensures area is positive regardless of vertex order.
**Use case 1**: Compute area without drawing anything.
**Use case 2**: Test for COLLINEARITY — three points are collinear iff the (signed) area expression equals 0.
**Example**: A(1, 1), B(2, 3), C(4, 5).
Area = 21∣1(3−5)+2(5−1)+4(1−3)∣=21∣−2+8−8∣=21·2 = 1.
Non-zero → not collinear.