3D Geometry
Midpoint of Line Segment
Grade 12
Question:
<p>Let the vertices of a triangle are A\((x_1, y_1, z_1)\), B\((x_2, y_2, z_2)\) and C\((x_3, y_3, z_3)\). The mid-points of sides AB, BC and CA are F(2, 3, −1), D(5, 7, 11) and E(0, 8, 5) respectively. Find \(x_1 + x_2\).</p>
Step-by-Step Solution
Key Concept: Use the midpoint formula for each side to set up equations relating the coordinates of the vertices.
Step 1: Since F is the midpoint of AB: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right) = (0, 8, 5)$ Step 2: From the x-coordinate: $\frac{x_1 + x_2}{2} = 0$ $x_1 + x_2 = 0$ ∴ Answer is 0
Correct Answer: 0