Question

Find the coordinate of the point of trisection (i.e. (-4,2) dividing the three equal parts) of the segment joining the points A(2,-2) and B(-7,4).

Answer 1

Hint: Trisection means dividing a line segment in three equal parts or dividing a line segment in the ratio 1:2 and 2:1 internally.In this question we have been given the ratio is 2:1. Start by considering 2 points which trisects the given line.

Let us assume two points $P$ and $Q$ to be the points of trisection.
Therefore, $AP = PQ = QB$
Let us assume $A\left( {2, - 2} \right) = A\left( {{x_1},{y_1}} \right)$ and $B\left( { - 7,4} \right) = B\left( {{x_2},{y_2}} \right)$
Now since it is given that it is a trisection therefore, that means $P$ will divide $AB$ in the ratio $1:2$ and $Q$ will divide $AB$ in the ratio $2:1$
Therefore, we are going to calculate the coordinates of $P$ using the section formula,
Coordinates of $P = \left( {\dfrac{{1 \times \left( { - 7} \right) + 2 \times 2}}{{1 + 2}},\dfrac{{1 \times 4 + 2 \times \left( { - 2} \right)}}{{1 + 2}}} \right) = \left( {\dfrac{{ - 7 + 4}}{3},\dfrac{{4 - 4}}{3}} \right)$

Note: Make sure you take the right values while assigning it in the formula.

If we solve it further, the point we get is,
                                = $\left( { - 1,0} \right)$
Similarly, we are going to calculate the coordinates of $Q$ using the section formula,
Coordinates of $Q = \left( {\dfrac{{2 \times \left( { - 7} \right) + 1 \times 2}}{{2 + 1}},\dfrac{{2 \times 4 + 1 \times \left( { - 2} \right)}}{{2 + 1}}} \right) = \left( {\dfrac{{ - 14 + 2}}{3},\dfrac{{8 - 2}}{3}} \right)$
If we solve it further, the point we get is,
                                = $\left( { - 4,2} \right)$
So, we have found the points $P$ and $Q$ as $\left( { - 1,0} \right)$ and $\left( { - 4,2} \right)$ respectively which are the points of intersection.