Question

The coordinates of the point(s), which trisect(s) the line segment joining the points $(1,-2)$ and $(-3,4),$ are

• A. $(0,\frac{-1}{2})$
• B. $(\frac{-1}{3},0)$
• C. $(-2,\frac{5}{2})$
• D. $(\frac{-5}{3},2)$

Answer 1

Answer: (B), (D)

$(\frac{-5}{3},2)$

Let $A\equiv (1,-2)$ and $B\equiv (-3,4).$
Let $P$ and $Q$ trisect the line segment $AB.$ Then
$\frac{AP}{PB}=\frac{1}{2}$
and $\frac{AQ}{QB}=\frac{2}{1}$


$P$ divides $AB$ internally in the ratio $1:2.$
$\therefore P\equiv (\frac{1(-3)+2\left(1\right)}{1+2},\frac{1\left(4\right)+2(-2)}{1+2})\Rightarrow P\equiv (\frac{-1}{3},0)$

$Q$ divides $AB$ internally in the ratio $2:1.$
$\therefore Q\equiv (\frac{2(-3)+1\left(1\right)}{2+1},\frac{2\left(4\right)+1(-2)}{2+1})\Rightarrow Q\equiv (\frac{-5}{3},2)$