Question 3
Determine if the points (1, 5), (2, 3) and (–2, –11) are collinear.
📋 Given Information
- First point: \( A(1, 5) \) where \( x_1 = 1, y_1 = 5 \)
- Second point: \( B(2, 3) \) where \( x_2 = 2, y_2 = 3 \)
- Third point: \( C(-2, -11) \) where \( x_3 = -2, y_3 = -11 \)
🎯 To Find
Whether the three given points lie on the same straight line (are collinear).
Definition: Three points A, B, and C are said to be collinear if they lie on the same straight line.
Methods to Check Collinearity:
- Area Method: If area of triangle = 0, points are collinear
- Slope Method: If slope AB = slope BC, points are collinear
- Distance Method: If AB + BC = AC (or similar), points are collinear
📐 Formula
📝 Step-by-Step Solution
Using A(1, 5), B(2, 3), C(-2, -11):
\[ \text{Area} = \frac{1}{2} |x_1(y_2 – y_3) + x_2(y_3 – y_1) + x_3(y_1 – y_2)| \] \[ = \frac{1}{2} |1(3 – (-11)) + 2((-11) – 5) + (-2)(5 – 3)| \]Calculate each part:
- \( 1(3 – (-11)) = 1 \times 14 = 14 \)
- \( 2((-11) – 5) = 2 \times (-16) = -32 \)
- \( (-2)(5 – 3) = (-2) \times 2 = -4 \)
Since Area = 11 ≠ 0, the three points are NOT collinear.
They form a triangle with area 11 square units.
🔍 Verification: Using Slope Method
Slope of AB:
\[ m_{AB} = \frac{y_2 – y_1}{x_2 – x_1} = \frac{3 – 5}{2 – 1} = \frac{-2}{1} = -2 \]Slope of BC:
\[ m_{BC} = \frac{y_3 – y_2}{x_3 – x_2} = \frac{-11 – 3}{-2 – 2} = \frac{-14}{-4} = \frac{7}{2} = 3.5 \]Conclusion: Since \( m_{AB} = -2 \neq 3.5 = m_{BC} \), the points do NOT lie on the same line.
💡 What if there’s a typo? Checking C(-2, 11) instead
If the third point were C(-2, 11) instead of C(-2, -11):
Using Area Method:
\[ \text{Area} = \frac{1}{2} |1(3 – 11) + 2(11 – 5) + (-2)(5 – 3)| \] \[ = \frac{1}{2} |-8 + 12 – 4| = \frac{1}{2} |0| = 0 \]✓ With C(-2, 11), the points WOULD be collinear!
Slope verification:
Slope AB = -2
Slope BC = \( \frac{11-3}{-2-2} = \frac{8}{-4} = -2 \) ✓
Both slopes are equal, confirming collinearity.
⚠️ Common Mistakes
❌ Mistake 1: Forgetting absolute value
✓ Correct: Always use \( |…| \) in the area formula
❌ Mistake 2: Sign errors with negative coordinates
Writing \( 3 – (-11) = -8 \) instead of 14
✓ Correct: \( 3 – (-11) = 3 + 11 = 14 \)
❌ Mistake 3: Calculation errors in subtraction
Writing \( 14 – 32 – 4 = -18 \) instead of -22
✓ Correct: \( 14 – 32 – 4 = -22 \)
💡 Key Points to Remember
- Area = 0: Points are collinear (lie on same line)
- Area ≠ 0: Points are NOT collinear (form a triangle)
- Slope Method: All pairs must have equal slopes for collinearity
- Sign Care: Be extra careful with negative coordinates
- Verification: Use multiple methods to confirm your answer
- Formula: \( \frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)| \)
📝 Practice Similar Problems
