Question

If the coordinates of mid-points of the sides of a triangle are \( (1, 5, -1), (0, 4, -2) \) and \( (2, 3, 4) \), then find the coordinates of its vertices.

Hint:

For finding the coordinates of triangle vertices from midpoints, use the midpoint formula and solve the system of equations.

Answer 1

Let the coordinates of the vertices of the triangle be \( A(x_1, y_1, z_1) \), \( B(x_2, y_2, z_2) \), and \( C(x_3, y_3, z_3) \). The midpoints of the sides of the triangle are given as: - Midpoint of \( BC \) is \( M_1 = (1, 5, -1) \),
- Midpoint of \( CA \) is \( M_2 = (0, 4, -2) \),
- Midpoint of \( AB \) is \( M_3 = (2, 3, 4) \).
Using the midpoint formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right). \] Now, solve the system of equations derived from these midpoints: \[ x_2 + x_3 = 2, \quad y_2 + y_3 = 10, \quad z_2 + z_3 = -2, \] \[ x_3 + x_1 = 0, \quad y_3 + y_1 = 8, \quad z_3 + z_1 = -4, \] \[ x_1 + x_2 = 4, \quad y_1 + y_2 = 6, \quad z_1 + z_2 = 8. \] Solving these equations gives the coordinates of the vertices as: \[ A(1, 2, 3), \quad B(3, 4, 5), \quad C(-1, 6, -7). \]
Final Answer: \[ \boxed{A(1, 2, 3), B(3, 4, 5), C(-1, 6, -7)}. \]