Question

Find the co-ordinates of the points of trisection of the line segment joining the points $(-2, 2)$ and $(7, -4)$.

Answer 1

The points divide the line segment into three equal parts, so the ratio of division is $1 : 2$ for the first point and $2 : 1$ for the second point.

For the first point, using the section formula, the coordinates dividing the segment $(-2, 2)$ and $(7, -4)$ in the ratio $1 : 2$ are:

$x = \frac{1 \times 7 + 2 \times (-2)}{1 + 2} = \frac{7 - 4}{3} = 1, \quad y = \frac{1 \times (-4) + 2 \times 2}{1 + 2} = \frac{-4 + 4}{3} = 0$

For the second point, dividing the segment in the ratio $2 : 1$, we get:

$x = \frac{2 \times 7 + 1 \times (-2)}{2 + 1} = \frac{14 - 2}{3} = 4, \quad y = \frac{2 \times (-4) + 1 \times 2}{2 + 1} = \frac{-8 + 2}{3} = -2$

So, the coordinates of the trisection points are:

$(1, 0) \quad \text{and} \quad (4, -2)$