Straight Lines
Collinearity of points
Grade 11
Question:
<p>The given points are <em>A = (0, 8/3)</em>, <em>B = (1, 3)</em>, <em>C = (82, 30)</em>. Then which of the following is true?</p>
<p>A, B, C form a triangle</p>
<p>A, B, C are collinear</p>
<p>A, B, C form a right-angled triangle</p>
<p>A, B, C form an equilateral triangle</p>
Step-by-Step Solution
Key Concept: Check if three points are collinear by verifying if the slope between any two pairs of points is identical. If slopes AB = slopes BC, the points lie on the same straight line.
<p><strong>Step 1:</strong> Find slope of AB</p><p>m₁ = (3 - 8/3)/(1 - 0) = (9/3 - 8/3)/1 = 1/3</p><p><strong>Step 2:</strong> Find slope of BC</p><p>m₂ = (30 - 3)/(82 - 1) = 27/81 = 1/3</p><p><strong>Step 3:</strong> Compare slopes</p><p>Since m₁ = m₂ = 1/3, and the slopes are equal, the three points A, B, C are collinear (they lie on the same straight line).</p><p><strong>Step 4:</strong> Verify with line equation</p><p>The line passing through A(0, 8/3) with slope 1/3: y = (1/3)x + 8/3</p><p>Check B(1, 3): y = 1/3(1) + 8/3 = 9/3 = 3 ✓</p><p>Check C(82, 30): y = 1/3(82) + 8/3 = 82/3 + 8/3 = 90/3 = 30 ✓</p><p>∴ Answer: The three points are collinear (Option B)</p>
Correct Answer: B