- A3
- B-6
- C-3
- D1
Step-by-Step Solution
Set the area to zero: → → → . Quick check: with , B(1, −3) is exactly the midpoint of AC, since . Adding and forgetting to halve gives −6; a sign slip when solving gives 3.
Answer: C) -3
If the points A(5, 1), B(1, ) and C(−3, −7) are collinear, find .
Class 10Coordinate GeometryGeometry — Coordinates
Key idea: Area of a triangle from vertices
Given A₁, ₁), B₂, ₂), C₃, ₃), the area is:
**Area = ₁₂ − ₃) + ₂₃ − ₁) + ₃₁ − ₂)|**
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 = ·2 = 1.
Non-zero → not collinear.
**Area = ₁₂ − ₃) + ₂₃ − ₁) + ₃₁ − ₂)|**
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 = ·2 = 1.
Non-zero → not collinear.
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