Geometry

Question Analysis:
This question asks us to identify the type of quadrilateral using coordinate geometry.
The main focus is checking whether the quadrilateral satisfies the conditions for a parallelogram:
Opposite sides are parallel (i.e., same slopes).
Alternatively, opposite sides can be equal in length (but here, we focus on slopes).

Key Concept Explanation:
A parallelogram is a quadrilateral where:
Opposite sides are parallel (i.e., have equal slopes).
Opposite sides are congruent (equal in length).
We'll start by checking the slopes of the sides, which tell us if the opposite sides are parallel.

Step-by-Step Solution:
Step 1: Compute the slopes of the sides
Slope of AB = (3 − 3)/(5 − 1) = 0/4 = 0
Slope of CD = (7 − 7)/(3 − 7) = 0/−4 = 0
Slope of BC = (7 − 3)/(7 − 5) = 4/2 = 2
Slope of DA = (7 − 3)/(3 − 1) = 4/2 = 2
Step 2: Verify opposite sides are parallel
AB ∥ CD ✔
BC ∥ DA ✔
Since both pairs of opposite sides are parallel, ABCD satisfies the definition of a parallelogram.
There are no two slopes where the product is -1, so this shape does not have right angles. 
Step 3: Optional — Check side lengths for further confirmation
AB = √[(5 − 1)² + (3 − 3)²] = √[16 + 0] = 4
BC = √[(7 − 5)² + (7 −...