Find the vertex C (x, y) of a ∆ABC whose other vertices are A(1, 5) and B(4, 7), and the centroid is D(4, 5).
C (7, 3)
The centroid of a triangle is the point where its three medians meet, and it divides each median in a 2:1 ratio. A key property is that its coordinates are simply the average of the three vertices' coordinates:
Centroid = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)
We are given A(1, 5), B(4, 7), the unknown vertex C(x, y), and the centroid D(4, 5). Setting each average equal to the corresponding centroid coordinate:
- x-coordinate: (1 + 4 + x)/3 = 4 → 5 + x = 12 → x = 7.
- y-coordinate: (5 + 7 + y)/3 = 5 → 12 + y = 15 → y = 3.
Therefore C = (7, 3).
A quick sanity check confirms it: (1 + 4 + 7)/3 = 12/3 = 4 and (5 + 7 + 3)/3 = 15/3 = 5, which matches D(4, 5) exactly. The option (7, 4) fails the y-check and (6, 3) fails the x-check, since swapping or mis-reading a coordinate no longer averages back to the given centroid. The correct answer is C (7, 3).
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