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)

$(-1,3,-2)$, $(-3,6,2)$

Answer 2

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 $Q\left(-3,6, 2\right)$.