P(3, 1) and R(-7, 5) are opposite vertices of a rhombus PQRS. What is the equation of the diagonal QS?

Introduction / Context:
This coordinate geometry problem involves a rhombus and its diagonals. We are given two opposite vertices of the rhombus and asked for the equation of the other diagonal. The key property used here is that the diagonals of any parallelogram, and therefore of a rhombus, bisect each other at a common midpoint.

Given Data / Assumptions:

• Rhombus PQRS has vertices P(3, 1) and R(-7, 5) as opposite vertices.
• Diagonal PR connects P and R.
• We must find the equation of diagonal QS.
• Diagonals of a rhombus intersect at their common midpoint.

Concept / Approach:
The diagonals PR and QS intersect at the same midpoint because a rhombus is a special type of parallelogram. We first compute the midpoint of PR. Any diagonal that passes through this midpoint is a candidate for QS. The options provide several lines; only one of them passes through the midpoint of PR. We do not need the exact slope of QS because checking which line passes through the midpoint is sufficient to identify the correct equation.

Step-by-Step Solution:
Step 1: Find the midpoint M of PR using the midpoint formula. Step 2: M = ((3 + (-7))/2, (1 + 5)/2) = (-4/2, 6/2) = (-2, 3). Step 3: Since diagonals of a rhombus bisect each other, diagonal QS must pass through M(-2, 3). Step 4: Substitute x = -2 and y = 3 into each option to see which equation is satisfied. Step 5: For 5x - 2y = 16, we get 5(-2) - 2(3) = -10 - 6 = -16, which is not 16. Step 6: For 5x - 2y = -16, we get -10 - 6 = -16, which satisfies the equation. Step 7: For the other options, substituting (-2, 3) does not satisfy their equations.

Verification / Alternative check:
Because diagonals of a parallelogram bisect each other, any diagonal must pass through the midpoint of the opposite diagonal. We have already confirmed that only 5x - 2y = -16 passes through M(-2, 3). This is sufficient to identify QS uniquely from the given choices without having to construct all vertices of the rhombus.

Why Other Options Are Wrong:

• Option A: 5x - 2y = 16 gives -16 on the left side at M, not 16.
• Option C: 5x + 2y = -16 gives -10 + 6 = -4, not -16.
• Option D: 5x + 2y = 16 gives -10 + 6 = -4, not 16.
• Option E: x + y = 1 gives -2 + 3 = 1, but there is no reason from rhombus properties to select this particular line over more specific options involving P and R.

Common Pitfalls:
Some learners mistakenly assume that the diagonals of any rhombus are always perpendicular, which is not necessarily true. Others may attempt to guess the equation by inspection rather than using the midpoint property. Computing the midpoint and checking each candidate equation systematically is the most reliable method.

Final Answer:
Therefore, the equation of diagonal QS is 5x - 2y = -16.