Question

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

Answer 1

Hint: Trisection of line means we have to divide the line into three parts. So first assume the coordinate of the two points between the two given points and then use the section formula to find the coordinate of these points.

Complete step-by-step answer:
In the question, it is given that a line is formed by joining the points (4,1) and (-2,-3).
And we have to find the coordinates of points which divide the line into three equal parts.
The diagram for question is given below:

Here R and S are the points which divide the line between P and Q into three parts.
Let the coordinates of point R be (${\text{(}}{{\text{x}}_1},{{\text{y}}_1})$ and that of S be $({{\text{x}}_2},{{\text{y}}_2})$.
According to section formula:
If the coordinates of two points be ${\text{(}}{{\text{x}}_1},{{\text{y}}_1})$and $({{\text{x}}_2},{{\text{y}}_2})$ and a third point(x,y) divides the line joining these two points in m:n ratio then:
x= $\dfrac{{{\text{m}}{{\text{x}}_2} + {\text{n}}{{\text{x}}_1}}}{{{\text{m + n}}}}$ , y= $\dfrac{{{\text{m}}{{\text{y}}_2} + {\text{n}}{{\text{y}}_1}}}{{{\text{m + n}}}}$ .
Now, we know that PR = RS = SQ.
$\therefore $ $\dfrac{{{\text{PR}}}}{{{\text{RQ}}}}$ =$\dfrac{1}{2}$ =$\dfrac{{\text{m}}}{{\text{n}}}$ .
Using the section formula for point R, we get:
${{\text{x}}_1} = \dfrac{{4 \times 2 + ( - 2) \times 1}}{{2 + 1}} = \dfrac{6}{3} = 2$.
${{\text{y}}_1} = \dfrac{{1 \times 2 + ( - 3 \times 1)}}{{1 + 2}} = \dfrac{-1}{3}$ .

Now for calculating the coordinates of point S, we have to calculate $\dfrac{{{\text{PS}}}}{{{\text{SQ}}}}$ .
PS =PR+RS and PR = RS = SQ.
$\therefore $ $\dfrac{{{\text{PS}}}}{{{\text{SQ}}}}$=$\dfrac{2}{1}$ .
 Using the section formula for point Q, we get:
$
  {{\text{x}}_2} = \dfrac{{4 \times 1 + ( - 2 \times 2)}}{{1 + 2}} = {\text{0}} \\
  {{\text{y}}_2} = \dfrac{{1 \times 1 + ( - 3 \times 2)}}{{1 + 2}} = \dfrac{{ - 5}}{3} \\
 $
Therefore, the trisection points of given points are R $(2, \dfrac{-1}{3} )$ and S $(0, \dfrac{{ - 5}}{3} )$.

Note: In this type of question, first of all mark the assumed points and then find the value of ratio in which the given unknown point divides the line then use the section formula . We should remember the section formula. We can also use the coordinates of points P and R or Points R and Q to find the coordinate of point S.