To determine the type of quadrilaterals, we need to examine the relationships between the sides and diagonals of quadrilateral [tex]\(ABCD\)[/tex] with vertices [tex]\( A(11, -7) \)[/tex], [tex]\( B(9, -4) \)[/tex], [tex]\( C(11, -1) \)[/tex], and [tex]\( D(13, -4) \)[/tex] and the modified quadrilateral [tex]\(ABCD'\)[/tex] where [tex]\( C'(11, 1) \)[/tex].
### Step 1: Check properties of Quadrilateral [tex]\(ABCD\)[/tex]
First, let's determine if quadrilateral [tex]\(ABCD\)[/tex] is a parallelogram, rectangle, or some other quadrilateral.
The primary properties we look for are:
- Opposite sides being equal for a parallelogram.
- Both the above property and the diagonals being equal for a rectangle.
Since we have calculated earlier, quadrilateral [tex]\(ABCD\)[/tex] satisfies neither of these properties (it is neither a parallelogram nor a rectangle).
### Conclusion for Quadrilateral [tex]\(ABCD\)[/tex]
Quadrilateral [tex]\(ABCD\)[/tex] is neither a parallelogram nor a rectangle.
### Step 2: Check properties of Quadrilateral [tex]\(ABC'D\)[/tex]
Now, shift vertex [tex]\(C\)[/tex] to [tex]\(C'(11, 1)\)[/tex] and form quadrilateral [tex]\(ABC'D\)[/tex]. Again, we need to check the properties:
- Opposite sides being equal for a parallelogram.
- Both the above property and the diagonals being equal for a rectangle.
From our calculations, quadrilateral [tex]\(ABC'D\)[/tex] also satisfies neither of these properties (it is neither a parallelogram nor a rectangle).
### Conclusion for Quadrilateral [tex]\(ABC'D\)[/tex]
Quadrilateral [tex]\(ABC'D\)[/tex] is neither a parallelogram nor a rectangle.
### Final Answer:
Quadrilateral [tex]\(ABCD\)[/tex] is a _general quadrilateral_ (not specifically a parallelogram or rectangle). Quadrilateral [tex]\(ABC'D\)[/tex] is also a _general quadrilateral_ (not specifically a parallelogram or rectangle).
