Question

Find the coordinates of the points that trisect the line segment AB joining the points \( A(-1, 2) \) and \( B(2, 8) \).

Hint:

The section formula can be used to find points that divide a line segment into any ratio, including trisection.

Answer 1

Step 1: Understand the concept of trisection.
Trisection means dividing the line segment into three equal parts. To find the coordinates of the points that trisect the line segment, we need to find the points that divide the segment into 1:2 and 2:1 ratios.
Step 2: Use the section formula.
The section formula for dividing a line segment in the ratio \( m:n \) is given by: \[ \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right) \]
Step 3: Apply the section formula for the two trisection points.
- For the point dividing the segment in the ratio 1:2 (closer to point A), let \( m = 1 \) and \( n = 2 \): \[ \left( \frac{1 \times 2 + 2 \times (-1)}{1+2}, \frac{1 \times 8 + 2 \times 2}{1+2} \right) = \left( \frac{2 - 2}{3}, \frac{8 + 4}{3} \right) = \left( 0, \frac{12}{3} \right) = (0, 4) \] - For the point dividing the segment in the ratio 2:1 (closer to point B), let \( m = 2 \) and \( n = 1 \): \[ \left( \frac{2 \times 2 + 1 \times (-1)}{2+1}, \frac{2 \times 8 + 1 \times 2}{2+1} \right) = \left( \frac{4 - 1}{3}, \frac{16 + 2}{3} \right) = \left( \frac{3}{3}, \frac{18}{3} \right) = (1, 6) \]
Step 4: Final coordinates.
The points that trisect the line segment are \( (0, 4) \) and \( (1, 6) \).