Find the coordinates of the points which divide the line segment joining A(−2, 2) and B(2, 8) into four equal parts.
\(\left(-1, \dfrac{7}{2}\right), (0, 5), \left(1, \dfrac{13}{2}\right)\)
We are given the endpoints of the line segment as \(A(-2, 2)\) and \(B(2, 8)\). We need three internal points that divide \(AB\) into four equal parts.
If the segment is divided into four equal parts, then the first point from \(A\) divides \(AB\) in the ratio \(1 : 3\), the second point in the ratio \(1 : 1\) (the midpoint), and the third point in the ratio \(3 : 1\).
Using the section formula, a point dividing the segment joining \((x_1, y_1)\) and \((x_2, y_2)\) in the ratio \(m : n\) internally has coordinates \(\left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right).\)
First point (ratio \(1 : 3\)) from \(A\): \(x = \frac{1\cdot 2 + 3\cdot (-2)}{1 + 3} = \frac{2 - 6}{4} = -1\), \(y = \frac{1\cdot 8 + 3\cdot 2}{4} = \frac{8 + 6}{4} = \frac{14}{4} = \frac{7}{2}\). So the first point is \(\left(-1, \frac{7}{2}\right).\)
Second point (midpoint, ratio \(1 : 1\)): \(x = \frac{-2 + 2}{2} = 0\), \(y = \frac{2 + 8}{2} = 5\). So the second point is \((0, 5).\)
Third point (ratio \(3 : 1\)) from \(A\): \(x = \frac{3\cdot 2 + 1\cdot (-2)}{3 + 1} = \frac{6 - 2}{4} = 1\), \(y = \frac{3\cdot 8 + 1\cdot 2}{4} = \frac{24 + 2}{4} = \frac{26}{4} = \frac{13}{2}\). So the third point is \(\left(1, \frac{13}{2}\right).\)
Therefore, the points which divide the segment joining \(A(-2, 2)\) and \(B(2, 8)\) into four equal parts are \(\left(-1, \frac{7}{2}\right), (0, 5), \left(1, \frac{13}{2}\right).\)