Question

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

Answer 1

Step 1: Understanding the problem:

We are given a line segment with endpoints \( (-2, 2) \) and \( (7, -4) \). The points divide the line segment into three equal parts. The first point divides the segment in the ratio \( 1 : 2 \), and the second point divides the segment in the ratio \( 2 : 1 \). We need to find the coordinates of these two points using the section formula.

Step 2: Using the section formula for the first point:

The section formula gives the coordinates of a point dividing a line segment in a given ratio. The formula is:
\[ x = \frac{m x_2 + n x_1}{m + n}, \quad y = \frac{m y_2 + n y_1}{m + n} \] where the coordinates of the endpoints are \( (x_1, y_1) \) and \( (x_2, y_2) \), and the point divides the segment in the ratio \( m : n \). For the first point, dividing the segment in the ratio \( 1 : 2 \), we have \( m = 1 \) and \( n = 2 \). The coordinates of the endpoints are \( (-2, 2) \) and \( (7, -4) \). Now, applying the section formula:
For \( x \)-coordinate: \[ x = \frac{1 \times 7 + 2 \times (-2)}{1 + 2} = \frac{7 - 4}{3} = 1 \] For \( y \)-coordinate: \[ y = \frac{1 \times (-4) + 2 \times 2}{1 + 2} = \frac{-4 + 4}{3} = 0 \] So, the coordinates of the first trisection point are \( (1, 0) \).

Step 3: Using the section formula for the second point:

For the second point, which divides the segment in the ratio \( 2 : 1 \), we have \( m = 2 \) and \( n = 1 \). Using the same endpoints \( (-2, 2) \) and \( (7, -4) \), we apply the section formula again:
For \( x \)-coordinate: \[ x = \frac{2 \times 7 + 1 \times (-2)}{2 + 1} = \frac{14 - 2}{3} = 4 \] For \( y \)-coordinate: \[ y = \frac{2 \times (-4) + 1 \times 2}{2 + 1} = \frac{-8 + 2}{3} = -2 \] So, the coordinates of the second trisection point are \( (4, -2) \).

Step 4: Conclusion:

The coordinates of the trisection points are \( (1, 0) \) and \( (4, -2) \).