ABC is a triangle with A(3,5). The mid-points of sides AB, AC are at (-1,2), (6,4) respectively. What are the coordinates of centroid of the triangle ABC?
- A. (\frac{8}{3},\frac{11}{3})
- B. (\frac{7}{3},\frac{7}{3}) ✓
- C. (2,\frac{8}{3})
- D. (\frac{8}{3},2)
Correct Answer: B. (\frac{7}{3},\frac{7}{3})
Explanation
Let B(x_1, y_1) and C(x_2, y_2). The midpoint of AB is (-1, 2), so \frac{x_1+3}{2} = -1 \implies x_1 = -5 and \frac{y_1+5}{2} = 2 \implies y_1 = -1. The midpoint of AC is (6, 4), so \frac{x_2+3}{2} = 6 \implies x_2 = 9 and \frac{y_2+5}{2} = 4 \implies y_2 = 3. The centroid G = (\frac{3-5+9}{3}, \frac{5-1+3}{3}) = (\frac{7}{3}, \frac{7}{3}).
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