3D Geometry
Distance from Point to Plane
Grade 12
Question:
<p>The length of the perpendicular from vertex D on the opposite face is</p>
<p>(a) <span class="math">\frac{14}{6}</span></p>
<p>(b) <span class="math">\frac{2}{6}</span></p>
<p>(c) <span class="math">\frac{3}{6}</span></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use the distance formula from a point to a plane defined by three points using the normal vector from cross product.
Given: Point G is \left(\frac{4}{3}, \frac{1}{3}, \frac{8}{3}\right) Step 1: Calculate |\mathbf{AG}|^2 = \left(\frac{5}{3}\right)^2 + \left(\frac{1}{9}\right)^2 + \left(\frac{5}{3}\right)^2 = \frac{51}{9} Step 2: Therefore, |\mathbf{AG}| = \frac{\sqrt{51}}{3} Step 3: Vectors on the base plane: \mathbf{AB} = -4\mathbf{i} + 4\mathbf{j}, \quad \mathbf{AC} = 2\mathbf{i} + 2\mathbf{j} + 2\mathbf{k} Step 4: The perpendicular distance from D(0, –5, 4) to the opposite face ABC is \frac{14}{6} ∴ Answer is (a) \frac{14}{6}
Correct Answer: A