Given Information:
The vertices of the quadrilateral are (1, 7), (4, 2), (1, 1), and (4, 4).

Explanation:
To determine the type of quadrilateral formed by these vertices, we need to consider the properties of different quadrilaterals.

Square:
A square is a quadrilateral with all sides of equal length and all angles equal to 90 degrees.

Parallelogram:
A parallelogram is a quadrilateral in which opposite sides are parallel.

Rhombus:
A rhombus is a quadrilateral with all sides of equal length.

Rectangle:
A rectangle is a quadrilateral with all angles equal to 90 degrees.

Using Properties:
To determine the type of quadrilateral, we can analyze the given vertices and their properties.

Side Lengths:
Using the distance formula, we can calculate the lengths of the sides of the quadrilateral:
AB = √[(4-1)^2 + (2-7)^2] = √[9 + 25] = √34
BC = √[(4-1)^2 + (4-2)^2] = √[9 + 4] = √13
CD = √[(1-4)^2 + (1-4)^2] = √[9 + 9] = √18
DA = √[(1-4)^2 + (1-7)^2] = √[9 + 36] = √45

Angle Measures:
Using the slope formula, we can calculate the slopes of the sides of the quadrilateral:
mAB = (2-7)/(4-1) = -5/3
mBC = (4-2)/(4-1) = 2/3
mCD = (1-4)/(1-4) = 3/0 (undefined)
mDA = (1-7)/(1-4) = 6/-3 = -2

Analysis:
- The quadrilateral does not have all sides of equal length, so it is not a square or a rhombus.
- The quadrilateral does not have all angles equal to 90 degrees, so it is not a rectangle.
- The quadrilateral does not have opposite sides that are parallel, so it is not a parallelogram.

Conclusion:
Based on the analysis above, the quadrilateral formed by the given vertices is not a square, parallelogram, rhombus, or rectangle. Therefore, the correct answer is option 'A' (None of the above).