To determine whether quadrilateral [tex]\(ABCD\)[/tex] is a rhombus, we need to check the properties of a rhombus. A rhombus is defined as a quadrilateral where all four sides are of equal length.
Let's calculate the lengths of the sides [tex]\(AB\)[/tex], [tex]\(BC\)[/tex], [tex]\(CD\)[/tex], and [tex]\(DA\)[/tex] using the distance formula. The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
First, we calculate the length of [tex]\(AB\)[/tex]:
- Coordinates of [tex]\(A\)[/tex] are [tex]\((6, 8)\)[/tex]
- Coordinates of [tex]\(B\)[/tex] are [tex]\((6, -5)\)[/tex]
[tex]\[
AB = \sqrt{(6 - 6)^2 + (-5 - 8)^2} = \sqrt{0 + (-13)^2} = \sqrt{169} = 13
\][/tex]
Next, we calculate the length of [tex]\(BC\)[/tex]:
- Coordinates of [tex]\(B\)[/tex] are [tex]\((6, -5)\)[/tex]
- Coordinates of [tex]\(C\)[/tex] are [tex]\((-5, -10)\)[/tex]
[tex]\[
BC = \sqrt{(-5 - 6)^2 + (-10 - (-5))^2} = \sqrt{(-11)^2 + (-5)^2} = \sqrt{121 + 25} = \sqrt{146} \approx 12.08
\][/tex]
Then, we calculate the length of [tex]\(CD\)[/tex]:
- Coordinates of [tex]\(C\)[/tex] are [tex]\((-5, -10)\)[/tex]
- Coordinates of [tex]\(D\)[/tex] are [tex]\((-5, 3)\)[/tex]
[tex]\[
CD = \sqrt{(-5 - (-5))^2 + (3 - (-10))^2} = \sqrt{0 + (13)^2} = \sqrt{169} = 13
\][/tex]
Finally, we calculate the length of [tex]\(DA\)[/tex]:
- Coordinates of [tex]\(D\)[/tex] are [tex]\((-5, 3)\)[/tex]
- Coordinates of [tex]\(A\)[/tex] are [tex]\((6, 8)\)[/tex]
[tex]\[
DA = \sqrt{(6 - (-5))^2 + (8 - 3)^2} = \sqrt{(11)^2 + (5)^2} = \sqrt{121 + 25} = \sqrt{146} \approx 12.08
\][/tex]
We observe that the sides [tex]\(AB\)[/tex] and [tex]\(CD\)[/tex] both measure 13 units, while [tex]\(BC\)[/tex] and [tex]\(DA\)[/tex] both measure approximately 12.08 units. Since the four sides are not all of equal length, quadrilateral [tex]\(ABCD\)[/tex] is not a rhombus.
Therefore, the correct answer is:
[tex]\[
\boxed{C} \text{ No, because } \overline{A B} \text{ is longer than } \overline{B C}.
\][/tex]
