<p>If a vertex of a triangle is (1, 1) and the mid-points of two sides through this vertex are (−1, 2) and (3, 2), then the centroid of the triangle is</p>
<p>\(\left(-1, \dfrac{7}{3}\right)\)</p>
<p>\(\left(-\dfrac{1}{3}, \dfrac{7}{3}\right)\)</p>
<p>\(\left(1, \dfrac{7}{3}\right)\)</p>
<p>\(\left(\dfrac{1}{3}, \dfrac{7}{3}\right)\)</p>
Step-by-Step Solution
Key Concept: Use the midpoint formula to find the other two vertices, then apply the centroid formula. The midpoint of a side gives us the average coordinates of its endpoints.
<p><strong>Step 1:</strong> Let the three vertices be A(1,1), B, and C. The midpoints of AB and AC are M₁(-1,2) and M₂(3,2) respectively.</p><p><strong>Step 2:</strong> Using midpoint formula, if B = (x₁, y₁), then midpoint of AB is ((1+x₁)/2, (1+y₁)/2) = (-1, 2)</p><p>This gives: (1+x₁)/2 = -1 ⟹ x₁ = -3 and (1+y₁)/2 = 2 ⟹ y₁ = 3</p><p>So B = (-3, 3)</p><p><strong>Step 3:</strong> Similarly, if C = (x₂, y₂), then ((1+x₂)/2, (1+y₂)/2) = (3, 2)</p><p>This gives: (1+x₂)/2 = 3 ⟹ x₂ = 5 and (1+y₂)/2 = 2 ⟹ y₂ = 3</p><p>So C = (5, 3)</p><p><strong>Step 4:</strong> The centroid G of triangle with vertices A(1,1), B(-3,3), C(5,3) is:</p><p>G = ((1-3+5)/3, (1+3+3)/3) = (3/3, 7/3) = (1, 7/3)</p><p>∴ Answer: C</p>
Correct Answer: C