The coordinates of the mid-point of the line joining the points (-3,5) and (2,10) is given by?

Analysis

To find the midpoint of a line segment joining two points $(x_1, y_1)$ and $(x_2, y_2)$, we use the midpoint formula. This formula calculates the average of the x-coordinates and the average of the y-coordinates.

Concept

The midpoint $M$ of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

Step-by-Step Solution

1. Identify the coordinates: • Point 1: $(x_1, y_1) = (-3, 5)$ • Point 2: $(x_2, y_2) = (2, 10)$

2. Apply the midpoint formula for the x-coordinate: $$x_m = \frac{x_1 + x_2}{2} = \frac{-3 + 2}{2} = \frac{-1}{2}$$

3. Apply the midpoint formula for the y-coordinate: $$y_m = \frac{y_1 + y_2}{2} = \frac{5 + 10}{2} = \frac{15}{2}$$

4. Combine the coordinates: The midpoint is $\left( \frac{-1}{2}, \frac{15}{2} \right)$.

Conclusion

Comparing our result to the given options, we find that it matches option D.

Exam Tip: When dealing with negative numbers in coordinate geometry, always be careful with the signs during addition. A common mistake is to subtract the numbers instead of adding them algebraically (e.g., $-3 + 2$ is $-1$, not $-5$). Always write out the signs clearly before calculating!