3D Geometry
Midpoint and Distance Formula
Grade 12
Question:
<p>ABC is a triangle with vertices A(0, 0, 6), B(0, 4, 0) and C(6, 0, 0). Let points D, E and F are the mid-points of BC, AC and AB, respectively. Find the length of median AD.</p>
Step-by-Step Solution
Key Concept: The median connects a vertex to the midpoint of the opposite side. Use the distance formula in 3D to find the length.
Step 1: Find the coordinates of midpoint D of BC. D = $\left(\frac{0+6}{2}, \frac{4+0}{2}, \frac{0+0}{2}\right) = (3, 2, 0)$ Step 2: Calculate the distance AD between A(0, 0, 6) and D(3, 2, 0). AD = $\sqrt{(0-3)^2 + (0-2)^2 + (6-0)^2}$ AD = $\sqrt{9 + 4 + 36} = \sqrt{49} = 7$ ∴ Length of median AD = 7
Correct Answer: 7