Question

(b)(ii) If the coordinates of midpoints of the sides of a triangle are: \[ (1, 5, -1), \quad (0, 4, -2), \quad (2, 3, 4), \] find the coordinates of its vertices.

Hint:

Midpoint formulas are useful for reconstructing triangle vertices from given midpoints.

Answer 1

Let the vertices of the triangle be \( A(x_1, y_1, z_1) \), \( B(x_2, y_2, z_2) \), \( C(x_3, y_3, z_3) \). The midpoints are: \[ M_1 = \left(\frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2}, \frac{z_2 + z_3}{2}\right), \quad M_2 = \left(\frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2}, \frac{z_1 + z_3}{2}\right), \] \[ M_3 = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right). \] Equating midpoints: \[ (1, 5, -1) = \left(\frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2}, \frac{z_2 + z_3}{2}\right), \quad \dots \] Solve for \( x_1, y_1, z_1, x_2, y_2, z_2, x_3, y_3, z_3 \).