Show that the line segment joining the points $(- 5, 8)$ and $(10, - 4)$ is trisected by the co-ordinate axes.

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Solution

Using the section formula, if a point $(x, y)$ divides the line joining the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ in the ratio $m : n$ , then
$(x, y) = (\frac{m x_{2} + n x_{1}}{m + n}, \frac{m y_{2} + n y_{1}}{m + n})$
Let the co-ordinate axes divides the line segment joining $(- 5, 8)$ and $(10, - 4)$ in the ratio $m : 1$
the point on the axes which is dividing the line segment will be $(\frac{1 (- 5) + m (10)}{m + 1}, \frac{1 (8) + m (- 4)}{m + 1}) = (\frac{10 m - 5}{m + 1}, \frac{- 4 m, + 8}{m + 1})$
If the axis is x-axis then $\frac{- 4 m + 8}{m + 1} = 0 ⟹ m = 2 ⟹ m : 1 = 2 : 1$
$⟹$ x-axis trisects the line segment.
If the axis is y-axis then $\frac{10 m - 5}{m + 1} = 0 ⟹ m = \frac{1}{2} ⟹ m : 1 = 1 : 2$
$⟹$ y-axis trisects the line segment

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Show that the line segment joining the points (−5,8) and (10,−4) is trisected by the co-ordinate axes.

Show that the line segment joining the points $(- 5, 8)$ and $(10, - 4)$ is trisected by the co-ordinate axes.