<p>If the coordinates of the middle points of the sides of a triangle are (1, 1), (2, -3) and (3, 4), find the vertices and the centroid of the triangle.</p>
Step-by-Step Solution
Key Concept: The midpoint of each side connects two vertices. Use the system of midpoint equations to express vertices in terms of the three given midpoints, then solve the resulting linear system to find coordinates.
<p><strong>Step 1:</strong> Let vertices be A(x₁, y₁), B(x₂, y₂), C(x₃, y₃). Given midpoints: D(1,1) of BC, E(2,-3) of CA, F(3,4) of AB.</p><p><strong>Step 2:</strong> Write midpoint equations:</p><p>• (x₂+x₃)/2 = 1, (y₂+y₃)/2 = 1 → x₂+x₃ = 2, y₂+y₃ = 2</p><p>• (x₃+x₁)/2 = 2, (y₃+y₁)/2 = -3 → x₃+x₁ = 4, y₃+y₁ = -6</p><p>• (x₁+x₂)/2 = 3, (y₁+y₂)/2 = 4 → x₁+x₂ = 6, y₁+y₂ = 8</p><p><strong>Step 3:</strong> Solve the system for x-coordinates: Adding all three equations: 2(x₁+x₂+x₃) = 12 → x₁+x₂+x₃ = 6</p><p>• x₁ = 6 - 2 = 4</p><p>• x₂ = 6 - 4 = 2</p><p>• x₃ = 6 - 6 = 0</p><p><strong>Step 4:</strong> Similarly for y-coordinates: 2(y₁+y₂+y₃) = 4 → y₁+y₂+y₃ = 2</p><p>• y₁ = 2 - 2 = 0</p><p>• y₂ = 2 - (-6) = 8</p><p>• y₃ = 2 - 8 = -6</p><p><strong>Step 5:</strong> Vertices are A(4, 0), B(2, 8), C(0, -6)</p><p><strong>Step 6:</strong> Centroid = ((4+2+0)/3, (0+8-6)/3) = (2, 2/3)</p><p><strong>Verification:</strong> Midpoint of BC = ((2+0)/2, (8-6)/2) = (1,1) ✓</p><p>∴ <strong>Vertices: (2, 8), (0, -6), (4, 0); Centroid: (2, 2/3)</strong></p>
Correct Answer: Vertices: (2, 8), (0, -6), (4, 0); Centroid: (2, 2/3)