Geometry

Question Analysis:
This question involves using slope and distance to classify a quadrilateral based on its coordinates.
The main focus is determining whether opposite sides are parallel and/or congruent.

Key Concept Explanation:
To determine the type of quadrilateral using coordinate proof, we can:
Use slope to check if opposite sides are parallel
Use distance to check if opposite sides are congruent
Use both to narrow down whether it is a parallelogram, rectangle, rhombus, or square

Step-by-Step Solution:
Step 1: Compute slopes of all sides
Slope of AB = (4 − 2)/(5 − 1) = 2/4 = 1/2
Slope of CD = (−2 − 0)/(3 − 7) = −2/−4 = 1/2
→ AB ∥ CD
Slope of BC = (0 − 4)/(7 − 5) = −4/2 = −2
Slope of DA = (2 − (−2))/(1 − 3) = 4/−2 = −2
→ BC ∥ DA
So both pairs of opposite sides are parallel.
And AB*BC = 1/2 * (-2) = -1, CD*DA = 1/2 * (-2) = -1
So AB and BC, CD and DA must be perpendicular.
Step 2: Compute lengths of all sides
AB = √[(5 − 1)² + (4 − 2)²] = √[16 + 4] = √20
CD = √[(3 − 7)² + (−2 − 0)²] = √[16 + 4] = √20
→ AB ≅ CD
BC = √[(7 − 5)²...