Geometry

Question Analysis:
This question involves identifying a quadrilateral using coordinate geometry.
The main focus is checking for equal side lengths (using the distance formula) and perpendicular diagonals (using slope), which define a rhombus.

Key Concept Explanation:
A rhombus is a special parallelogram where:
All four sides are congruent (equal in length).
The diagonals are perpendicular (form right angles).
To prove it's a rhombus:
Use the distance formula for all sides.
Use the slope formula to check if diagonals are perpendicular.

Step-by-Step Solution:
Step 1: Check side lengths
PQ: √[(4−1)² + (6−2)²] = √[9 + 16] = √25 = 5
QR: √[(7−4)² + (2−6)²] = √[9 + 16] = √25 = 5
RS: √[(4−7)² + (−2−2)²] = √[9 + 16] = √25 = 5
SP: √[(1−4)² + (2−(−2))²] = √[9 + 16] = √25 = 5
→ All sides = 5 → PQRS has four congruent sides ✔
Step 2: Check if diagonals are perpendicular
Diagonal PR: from P(1, 2) to R(7, 2) → slope = (2−2)/(7−1) = 0/6 = 0
Diagonal QS: from Q(4, 6) to S(4, −2) → slope = (−2−6)/(4−4) = −8/0 → undefined
→ One diagonal has slope 0 (horizontal), the other is undefined (vertical)
→ Their slopes are negative reciprocals (0 × undefined is conceptually "perpendicular") ✔
So the diagonals are perpendicular, and all four sides are equal.
Step 3: Check slopes of all sides
PQ: P(1, 2) to Q(4, 6) → slope = (6−2)/(4−1) = 4/3
QR: Q(4, 6) to R(7, 2) → slope = (2−6)/(7−4) = -4/3
RS: P(7, 2) to S(4, -2) → slope = (-2−2)/(4−7) = 4/3
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