In coordinate geometry, use the midpoint formula to find the coordinates of the midpoint of the line segment joining the points C(3, -5) and D(-7, 3).

Introduction / Context:
This problem tests understanding of the midpoint formula in coordinate geometry. The midpoint of a line segment gives the point that lies exactly halfway between two endpoints in the Cartesian plane. This concept is widely used in analytic geometry, physics, and many aptitude questions that involve distances and positions.

Given Data / Assumptions:

• Endpoint C has coordinates (3, -5).
• Endpoint D has coordinates (-7, 3).
• We assume a standard Cartesian coordinate system with x and y axes.
• We need the midpoint of the segment joining C and D.

Concept / Approach:
The midpoint formula states that for two points (x1, y1) and (x2, y2), the midpoint M has coordinates ((x1 + x2)/2, (y1 + y2)/2). We simply substitute the given coordinates of C and D into this formula. It is important to add the x coordinates together and divide by 2, and then add the y coordinates together and divide by 2, keeping the signs correct.

Step-by-Step Solution:
Label the points: C(3, -5) so x1 = 3, y1 = -5; D(-7, 3) so x2 = -7, y2 = 3.Apply the midpoint formula for the x coordinate: (x1 + x2)/2 = (3 + (-7))/2 = (3 - 7)/2 = -4/2 = -2.Apply the midpoint formula for the y coordinate: (y1 + y2)/2 = (-5 + 3)/2 = (-2)/2 = -1.Thus, the midpoint M has coordinates (-2, -1).Therefore, the required midpoint of the line segment joining C and D is (-2, -1).

Verification / Alternative check:
We can verify visually by checking that the midpoint is equally distant from both endpoints. The horizontal change from C to M is from x = 3 to x = -2, a difference of -5 units. From M to D, the horizontal change is from x = -2 to x = -7, which is also -5 units. The vertical change from C to M is from y = -5 to y = -1, a difference of 4 units. From M to D, the vertical change is from y = -1 to y = 3, also 4 units. The equal changes confirm that M is indeed the midpoint.

Why Other Options Are Wrong:

• (5, -4) does not lie halfway between C and D and does not satisfy the midpoint formula.
• (-5, 4) reverses both signs and is not centered between the two given points.
• (2, 1) lies closer to C and does not balance the distances to D.
• (0, 0) is the origin, which is not halfway between C(3, -5) and D(-7, 3).

Common Pitfalls:

• Forgetting to divide by 2 after adding coordinates.
• Making sign errors when adding negative numbers, especially -7 and -5.
• Swapping x and y coordinates by mistake.

Final Answer:
(-2, -1)