3D Geometry
Tetrahedron
Grade 12
Question:
<p>Let <i>A</i>(<i>x</i><sub>1</sub>, <i>y</i><sub>1</sub>, <i>z</i><sub>1</sub>), <i>B</i>(<i>x</i><sub>2</sub>, <i>y</i><sub>2</sub>, <i>z</i><sub>2</sub>), <i>C</i>(<i>x</i><sub>3</sub>, <i>y</i><sub>3</sub>, <i>z</i><sub>3</sub>), <i>D</i>(<i>x</i><sub>4</sub>, <i>y</i><sub>4</sub>, <i>z</i><sub>4</sub>) be the vertices of a tetrahedron. If <i>E</i> is the centroid of face <i>BCD</i> and <i>G</i> is the centroid of <i>ABCD</i>, then find the value of <i>K</i> such that <i>AG</i> = <i>K</i>(<i>AE</i>).</p>
Step-by-Step Solution
Key Concept: The centroid of a tetrahedron divides the line segment from any vertex to the centroid of the opposite face in the ratio 3:1.
Solution: Using the property that the centroid divides the median from a vertex to the centroid of the opposite face in the ratio 3:1, we have: AG = (3/4)( AE ) ∴ K = 3/4
Correct Answer: 3/4