Given the vertices [tex]\(A(11,-7), B(9,-4), C(11,-1)\)[/tex], and [tex]\(D(13,-4)\)[/tex], and comparing them to the provided answers, we proceed as follows:
First, we determine the lengths of the sides of the quadrilateral ABCD.
For [tex]\(AB\)[/tex]:
[tex]\[
AB = \sqrt{(9 - 11)^2 + (-4 - (-7))^2} = \sqrt{(-2)^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.605551275463989
\][/tex]
For [tex]\(BC\)[/tex]:
[tex]\[
BC = \sqrt{(11 - 9)^2 + (-1 - (-4))^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.605551275463989
\][/tex]
For [tex]\(CD\)[/tex]:
[tex]\[
CD = \sqrt{(13 - 11)^2 + (-4 - (-1))^2} = \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.605551275463989
\][/tex]
For [tex]\(DA\)[/tex]:
[tex]\[
DA = \sqrt{(13 - 11)^2 + (-4 - (-7))^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.605551275463989
\][/tex]
All sides of quadrilateral [tex]\(ABCD\)[/tex] are equal to [tex]\(\sqrt{13} \approx 3.605551275463989\)[/tex].
Next, we need to determine if this is a parallelogram. For it to be a parallelogram, opposite sides must be equal and parallel.
Additionally, we check the diagonals to see if they're equal:
[tex]\[
AC = \sqrt{(11 - 11)^2 + (-1 - (-7))^2} = \sqrt{0 + 36} = 6
\][/tex]
[tex]\[
BD = \sqrt{(13 - 9)^2 + (-4 - (-4))^2} = \sqrt{4^2 + 0} = 4
\][/tex]
Since AC and BD do not equal, ABCD is not a parallelogram; however, as all sides are equal and its diagonals do not bisect each other equally, it is a rhombus.
Next, we consider [tex]\(C^{\prime}(11, 1)\)[/tex] instead of [tex]\(C\)[/tex]. We compute the distances:
For [tex]\(B C'\)[/tex]:
[tex]\[
BC' = \sqrt{(11 - 9)^2 + (1 - (-4))^2} = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.385164807134504
\][/tex]
For [tex]\(C' D\)[/tex]:
[tex]\[
C'D = \sqrt{(11 - 13)^2 + (1 - (-4))^2} = \sqrt{(-2)^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.385164807134504
\][/tex]
For [tex]\(DA\)[/tex]:
[tex]\[
DA = \sqrt{(13 - 11)^2 + (-4 - (-7))^2} = \sqrt{2^2 + 3^2} = \sqrt{13} \approx 3.605551275463989
\][/tex]
Given the difference in side lengths [tex]\( \sqrt{13}\)[/tex] and [tex]\( \approx \sqrt{29}\)[/tex], [tex]\(AB' C' D\)[/tex] cannot be a parallelogram or a rhombus. Hence, it’s a general quadrilateral, not fitting any special categories of quadrilaterals listed here.
Therefore:
Quadrilateral [tex]\(ABCD\)[/tex] is a rhombus. Quadrilateral [tex]\(AB'C'D\)[/tex] would be a quadrilateral.
Final Answer:
Quadrilateral [tex]\(ABCD\)[/tex] is a [tex]\(\boxed{\text{Rhombus}}\)[/tex] and [tex]\(AB'C'D\)[/tex] would be a [tex]\(\boxed{\text{Quadrilateral}}\)[/tex].
