3D Geometry
Centroid Distance from Line Intersection
nta_pyq_2024_jan
Grade 12
Question:
Let $PQR$ be a triangle with $R(-1,4,2)$. Suppose $M(2,1,2)$ is the midpoint of $PQ$. The distance of the centroid of $\triangle PQR$ from the point of intersection of the lines $\dfrac{x-2}{0}=\dfrac{y}{2}=\dfrac{z+3}{-1}$ and $\dfrac{x-1}{1}=\dfrac{y+3}{-3}=\dfrac{z+1}{1}$ is
69
9
$\sqrt{69}$
$\sqrt{99}$
Step-by-Step Solution
Key Concept: Centroid $G=\frac{P+Q+R}{3}=\frac{2M+R}{3}=\frac{(4+(-1),2+4,4+2)}{3}=(1,2,2)$. Find intersection of the two lines and compute distance from $G$.
Centroid $(1,2,2)$, intersection $(2,-6,0)$. Distance $=\sqrt{69}$.
Correct Answer: 3