Question

The coordinate of the points which trisect the line segment joining the points $A(2,1,-3)$ and $B(5,-8,3)$, are

• A. $(3,-2,-1)$ and $(4,-5,1)$
• B. $(3,2,1)$ and $(4,5,2)$
• C. $(-3,5,1)$ and $(3,4,-5)$
• D. None of the above

Answer 1

Answer: (A)

Let $R_1$ and $R_2$ are the points with coordinates $\left(x_1,y_1,z_1\right)$ and $\left(x_2,y_2,z_3\right)$ which trisect the line segment $AB$.

Now, $A{R}_1:R_{1}B=1:2$
$\Rightarrow x_1=\frac{1\times 5+2\times 2}{1+2}=\frac{9}{3}=3$
$y_1=\frac{1\times (-8)+2\times 1}{1+2}=\frac{-6}{3}=-2$
and $z_1=\frac{1\times 3+2\times -3}{1+2}=\frac{-3}{3}=-1$
$\therefore R_1=(3,-2,-1)$
Similarly, $A{R}_2:R_{2}B=2:1$
$\Rightarrow x_2=\frac{2\times 5+1\times 2}{1+2}=\frac{12}{3}=4$
and $y_2=\frac{2\times (-8)+1\times 1}{1+2}=\frac{-15}{3}=-5$
$z_2=\frac{2\times 3+1\times (-3)}{1+2}=\frac{3}{3}=1$
$\therefore$ Coordinates of $R_2$ are $(4,-5,1)$.