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A quadrilateral has vertices [tex]\( A(11,-7), B(9,-4), C(11,-1), \)[/tex] and [tex]\( D(13,-4) \)[/tex].
Quadrilateral [tex]\( ABCD \)[/tex] is a [tex]\(\square\)[/tex] [Choose from drop-down].
If the vertex [tex]\( C(11,-1) \)[/tex] were shifted to the point [tex]\( C^{\prime}(11,1) \)[/tex], quadrilateral [tex]\( ABC^{\prime}D \)[/tex] would be a [tex]\(\square\)[/tex] [Choose from drop-down].

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To determine the types of quadrilaterals given the vertices, follow these steps:

1. Identify the vertices of the quadrilateral [tex]\(A B C D\)[/tex]:
- [tex]\(A(11, -7)\)[/tex]
- [tex]\(B(9, -4)\)[/tex]
- [tex]\(C(11, -1)\)[/tex]
- [tex]\(D(13, -4)\)[/tex]

2. Calculate the distances between the vertices to find the lengths of the sides and diagonals:
- [tex]\(AB\)[/tex]
- [tex]\(BC\)[/tex]
- [tex]\(CD\)[/tex]
- [tex]\(DA\)[/tex]
- [tex]\(AC\)[/tex]
- [tex]\(BD\)[/tex]

3. Classify Quadrilateral [tex]\(A B C D\)[/tex]:
- Determine if [tex]\(AB = BC = CD = DA\)[/tex]. If true, it's a square or rhombus.
- Determine if [tex]\(AB = CD\)[/tex] and [tex]\(BC = DA\)[/tex]. If true, it's a rectangle or parallelogram.
- If neither condition is met, it's other type of quadrilateral.

4. Shift vertex [tex]\(C(11, -1)\)[/tex] to [tex]\(C'(11, 1)\)[/tex] and denote the new vertex as [tex]\(C'\)[/tex].

5. Classify the new Quadrilateral [tex]\(A B C' D\)[/tex]:
- Calculate the new distances [tex]\(BC'\)[/tex], [tex]\(C'D\)[/tex], and [tex]\(AC'\)[/tex].
- Determine if [tex]\(AB = BC' = C'D = DA\)[/tex]. If true, it's a square or rhombus.
- Determine if [tex]\(AB = C'D\)[/tex] and [tex]\(BC' = DA\)[/tex]. If true, it's a rectangle or parallelogram.
- If neither condition is met, it's other type of quadrilateral.

Based on these calculations:
- Quadrilateral [tex]\(A B C D\)[/tex] is a square or rhombus.
- Quadrilateral [tex]\(A B C' D\)[/tex] changes and is other type of quadrilateral.

Therefore, fill in the drop-down menus as follows:

Quadrilateral [tex]\(A B C D\)[/tex] is a square or rhombus. If the vertex [tex]\(C(11,-1)\)[/tex] were shifted to the point [tex]\(C'(11,1)\)[/tex], quadrilateral [tex]\(A B C' D\)[/tex] would be another type of quadrilateral.
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