Question

Find the coordinates of the points of trisection of the line segment joining $(4,-1)$ and $(-2,-3)$.

Answer 1

Given coordinates are$(x_1,y_1)=$$(4,-1)$

and $(x_2,y_2)=(-2,-3)$

Means of trisection:- A line segment in three equal parts then ratio is $1:2$ and $2:1$ internally.

Case (1) If $m_1:m_2=1:2$

Then, using formula

$(x,y)=(\frac{m_1{x}_2+m_2{x}_1}{m_1+m_2},\frac{m_1{y}_2+m_2{y}_1}{m_1+m_2})$

$(x,y)=(\frac{1\times (-2)+2\times 4}{1+2},\frac{1\times (-3)+2\times (-1)}{1+2})$

$(x,y)=(\frac{-2+8}{3},\frac{-3-2}{3})$

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

Case (2):-

If $m_1:m_2=2:1$

Then, using formula

$(x,y)=(\frac{m_1{x}_2+m_2{x}_1}{m_1+m_2},\frac{m_1{y}_2+m_2{y}_1}{m_1+m_2})$

$(x,y)=(\frac{2\times (-2)+1\times 4}{2+1},\frac{2\times (-3)+1\times (-1)}{1+2})$

$(x,y)=(\frac{-4+4}{3},\frac{-6-1}{3})$

$(x,y)=(0,-\frac{7}{3})$

Hence, this is the answer.