Question

Find the coordinates of the points which divide the line segment \( AB \) joining points \( A (-2, 2) \) and \( B (2, 8) \) into four equal parts.

Hint:

To divide a line segment into equal parts, use the section formula by assigning the appropriate ratio for the divisions.

Answer 1

The coordinates of the points dividing the line segment into four equal parts are found using the section formula. If the line segment \( AB \) is divided in the ratio \( 1:3 \), \( 2:2 \), and \( 3:1 \), we find the coordinates as follows: 1. For \( P_1 \) (Dividing the segment in the ratio \( 1:3 \)): The coordinates of \( P_1 \) are given by the section formula: \[ P_1 = \left( \frac{3x_1 + 1x_2}{1+3}, \frac{3y_1 + 1y_2}{1+3} \right), \] where \( A(x_1, y_1) = (-2, 2) \) and \( B(x_2, y_2) = (2, 8) \). Substituting the values: \[ P_1 = \left( \frac{3(-2) + 1(2)}{4}, \frac{3(2) + 1(8)}{4} \right) = \left( \frac{-6 + 2}{4}, \frac{6 + 8}{4} \right) = \left( \frac{-4}{4}, \frac{14}{4} \right) = (-1, 3.5). \] 2. For \( P_2 \) (Dividing the segment in the ratio \( 2:2 \)): The coordinates of \( P_2 \) are: \[ P_2 = \left( \frac{2x_1 + 2x_2}{2+2}, \frac{2y_1 + 2y_2}{2+2} \right) = \left( \frac{2(-2) + 2(2)}{4}, \frac{2(2) + 2(8)}{4} \right) = \left( \frac{-4 + 4}{4}, \frac{4 + 16}{4} \right) = (0, 5). \] 3. For \( P_3 \) (Dividing the segment in the ratio \( 3:1 \)): The coordinates of \( P_3 \) are: \[ P_3 = \left( \frac{1x_1 + 3x_2}{1+3}, \frac{1y_1 + 3y_2}{1+3} \right) = \left( \frac{1(-2) + 3(2)}{4}, \frac{1(2) + 3(8)}{4} \right) = \left( \frac{-2 + 6}{4}, \frac{2 + 24}{4} \right) = \left( \frac{4}{4}, \frac{26}{4} \right) = (1, 6.5) \] Thus, the points dividing the line segment into four equal parts are \( P_1(-1, 3.5) \), \( P_2(0, 5) \), and \( P_3(1, 6.5) \).
Conclusion: The coordinates of the points which divide the line segment \( AB \) into four equal parts are \( P_1(-1, 3.5) \), \( P_2(0, 5) \), and \( P_3(1, 6.5) \).