Let's denote the vertices of the triangle as A, B, and C. The given midpoints are for the sides AB, BC, and CA. The coordinates of the midpoints are as follows:
Midpoint of AB: (2, 4)
Midpoint of BC: (1/2, 1/7)
Midpoint of CA: (5/2, 9/2)
Now, let's find the coordinates of the vertices:
1. Coordinates of A: Twice the x-coordinate of the midpoint of BC minus the x-coordinate of the midpoint of CA, and similarly for the y-coordinate.
\(A = (2 \times \frac{1}{2} - \frac{5}{2}, 2 \times \frac{1}{7} - \frac{9}{2})\)
\(A = (-\frac{4}{2}, -\frac{18}{7})\)
\(A = (-2, -\frac{18}{7})\)
2. Coordinates of B: Twice the x-coordinate of the midpoint of CA minus the x-coordinate of the midpoint of AB, and similarly for the y-coordinate.
\(B = (2 \times \frac{5}{2} - 2, 2 \times \frac{9}{2} - 4)\)
\(B = (\frac{5}{2}, 9 - 4)\)
\(B = (\frac{5}{2}, 5)\)
3. Coordinates of C: Twice the x-coordinate of the midpoint of AB minus the x-coordinate of the midpoint of BC, and similarly for the y-coordinate.
\(C = (2 \times 2 - \frac{1}{2}, 2 \times 4 - \frac{1}{7})\)
\(C = (\frac{4}{2}, 8 - \frac{1}{7})\)
\(C = (2, \frac{55}{7})\)
Therefore, the coordinates of the vertices A, B, and C are:
\(A = (-2, -\frac{18}{7})\)
\(B = (\frac{5}{2}, 5)\)
\(C = (2, \frac{55}{7})\)