Straight Lines
Centroid and collinearity
Grade 11
Question:
<p>If <span>\((ax_1 + by_1 + c) + (ax_2 + by_2 + c) + (ax_3 + by_3 + c) = 0\)</span>, show that the line <span>\(ax + by + c = 0\)</span> passes through the centroid of triangle with vertices <span>\((x_1,y_1), (x_2,y_2), (x_3,y_3)\)</span>. Find the value <span>\(15\)</span> times the number of such conditions.</p>
Step-by-Step Solution
Key Concept: The given condition states that the sum of the line's values at three vertices equals zero, which is equivalent to saying the line passes through the centroid (since the centroid's coordinates are the average of the vertices' coordinates). This is because ax + by + c evaluated at the centroid equals the average of its values at the three vertices.
<p><strong>Step 1:</strong> Given condition: (ax₁ + by₁ + c) + (ax₂ + by₂ + c) + (ax₃ + by₃ + c) = 0</p><p><strong>Step 2:</strong> Rewrite as: a(x₁ + x₂ + x₃) + b(y₁ + y₂ + y₃) + 3c = 0</p><p><strong>Step 3:</strong> Divide by 3: a·(x₁ + x₂ + x₃)/3 + b·(y₁ + y₂ + y₃)/3 + c = 0</p><p><strong>Step 4:</strong> The centroid G has coordinates ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)</p><p><strong>Step 5:</strong> The equation from Step 3 shows that a(x_G) + b(y_G) + c = 0, meaning the line ax + by + c = 0 passes through the centroid.</p><p><strong>Step 6:</strong> The problem asks for 15 times the number of such conditions. There is exactly ONE such geometric condition (the line passing through the centroid).</p><p>∴ Answer: 15 × 1 = <strong>15</strong></p>
Correct Answer: 15