25. If \( \left(\frac{3}{2}, 0\right),\left(\frac{-3}{2}, 6\right) \) and \( (-1,6) \) are vertices of a triangle and centroid is \( \left(\frac{-1}{3}, k\right) \), then \( \mathrm{k}= \)

To find the value of \( k \), we first need to calculate the centroid of the triangle formed by the points \( \left(\frac{3}{2}, 0\right) \), \( \left(\frac{-3}{2}, 6\right) \), and \( (-1, 6) \). The formula for the centroid \( (G_x, G_y) \) is given by: \[ G_x = \frac{x_1 + x_2 + x_3}{3}, \quad G_y = \frac{y_1 + y_2 + y_3}{3} \] Substituting the coordinates of the vertices: 1. \( x_1 = \frac{3}{2}, y_1 = 0 \) 2. \( x_2 = \frac{-3}{2}, y_2 = 6 \) 3. \( x_3 = -1, y_3 = 6 \) Now calculating \( G_x \): \[ G_x = \frac{\frac{3}{2} + \frac{-3}{2} + (-1)}{3} = \frac{0 - 1}{3} = \frac{-1}{3} \] Next, we calculate \( G_y \): \[ G_y = \frac{0 + 6 + 6}{3} = \frac{12}{3} = 4 \] So we have the centroid \( G \) as \( \left(\frac{-1}{3}, 4\right) \). Since we know that the centroid is given as \( \left(\frac{-1}{3}, k\right) \), we can determine: \[ k = 4 \] Thus, \( k = 4 \).