P(4, 2) and R(−2, 0) are opposite vertices of a rhombus PQRS. What is the equation of the diagonal QS?

Introduction / Context:
This question tests a key property of rhombuses and parallelograms in coordinate geometry: the behavior of their diagonals. In a rhombus, opposite vertices are joined by diagonals that bisect each other, and in many exam questions we also use the fact that the diagonals can be taken as perpendicular. Here, we are given two opposite vertices and asked for the equation of the other diagonal.

Given Data / Assumptions:

- Rhombus PQRS has opposite vertices P(4, 2) and R(−2, 0).
- PR and QS are the diagonals of the rhombus.
- Diagonals of a rhombus bisect each other at their common midpoint.
- The diagonal QS passes through that midpoint and is perpendicular to PR in this configuration.

Concept / Approach:
The strategy is to find the equation of PR, determine its slope, and then use perpendicularity to obtain the slope of QS. Next, we find the midpoint of PR; this point is also the midpoint of QS. Finally, using the slope of QS and the midpoint, we write the equation of QS and match it with the given options.

Step-by-Step Solution:
Step 1: Compute the midpoint M of PR. P is (4, 2) and R is (−2, 0). The midpoint is M = ((4 + (−2))/2, (2 + 0)/2) = (2/2, 2/2) = (1, 1).Step 2: Find the slope of PR. The slope m_PR is (0 − 2) / (−2 − 4) = −2 / −6 = 1/3.Step 3: Since QS is taken to be perpendicular to PR, its slope m_QS is the negative reciprocal of 1/3, which is −3.Step 4: Use the point slope form with slope −3 and point M(1, 1). The equation is y − 1 = −3(x − 1).Step 5: Expand: y − 1 = −3x + 3, so y = −3x + 4. Rearranging, we get 3x + y = 4.

Verification / Alternative check:
Check if the midpoint M(1, 1) lies on the line 3x + y = 4. Substituting x = 1 and y = 1, we get 3 * 1 + 1 = 4, which is true. The slope of this line is −3, which is indeed perpendicular to the slope 1/3 of PR. Thus, the line 3x + y = 4 is a consistent equation for diagonal QS.

Why Other Options Are Wrong:
- x − 3y = −2 and x − 3y = 2 both have slope 1/3, which makes them parallel to PR rather than perpendicular.
- 3x + y = −4 has slope −3 but does not pass through the midpoint (1, 1), since 3 * 1 + 1 = 4, not −4.
- y = −3x + 1 has the correct slope but passes through (0, 1) instead of (1, 1), so it is not the correct diagonal.

Common Pitfalls:
Students may forget that diagonals of a rhombus (and parallelogram) intersect at their common midpoint or confuse parallel and perpendicular slopes. Always compute the midpoint of the known diagonal and use that point along with the slope to determine the correct equation of the other diagonal.

Final Answer:
The equation of diagonal QS is 3x + y = 4.