Question

Find the points of trisection of the segment joining the points $A(1,0,-6)$ and $B(-5,9,6)$.

• A. $(-1,3,-2)$, $(-3,6,2)$
• B. $(1,-3,-2)$, $(3,-6,2)$
• C. $(1,3,2)$, $(3,6,2)$
• D. $(-1,3,-2)$, $(-3,-6,2)$

Answer 1

Answer: (A)

Let $P$ and $Q$ be the points of trisection of the segment $[AB]$, then $P$ divides $[AB]$ in the ratio $1:2$ and $Q$ divides $[AB]$ in the ratio $2:1$.
$\therefore P\equiv \left(\frac{1\times \left(-5\right)+2\times 1}{1+2},\frac{1\times 9+2\times 0}{1+2},\frac{1\times 6+2\times \left(-6\right)}{1+2}\right)$,
i.e., $P=\left(-1,3,-2\right)$
and $Q\equiv \left(\frac{2\times \left(-5\right)+1\times 1}{2+1},\frac{2\times 9+1\times 0}{2+1},\frac{2\times 6+1\times \left(-6\right)}{2+1}\right)$,
i.e., $Q$ is $\left(-3,6,2\right)$
Hence, the required points of trisection are $P\left(-1,3,-2\right)$ and $Q\left(-3,6,2\right)$.