If P(2, m) is the midpoint of the line joining Q(m, n) and R(n, -4), find the values of m and n.

Analysis

To solve this problem, we use the Midpoint Formula. If a point $P(x, y)$ is the midpoint of a line segment joining two points $Q(x_1, y_1)$ and $R(x_2, y_2)$, the coordinates of the midpoint are given by:

Concept

In this question, we are given:

• Midpoint $P = (2, m)$
• Point $Q = (m, n)$
• Point $R = (n, -4)$

By substituting these values into the midpoint formula, we create a system of two equations that we can solve simultaneously.

Step-by-Step Solution

Step 1: Set up the equations for the x and y coordinates. For the x-coordinate:

Multiply both sides by 2:

For the y-coordinate:

Multiply both sides by 2:

Step 2: Solve the system of equations. From Equation 1, we can express $n$ in terms of $m$:

Substitute this into Equation 2:

Step 3: Find the value of n. Substitute $m = 0$ back into Equation 1:

Thus, $m = 0$ and $n = 4$.

Conclusion

Comparing our result ($m = 0, n = 4$) with the given options, it matches option A.

Exam Tip

When dealing with midpoint questions, always write out your $x_1, y_1, x_2, y_2$ clearly before plugging them into the formula. This prevents "sign errors" (like forgetting the negative sign in $-4$), which is the most common reason students lose marks in coordinate geometry.