Quadrilateral \( A B C D \) has the following vertices: - \( A(6,8) \) - \( B(6,-5) \) - \( C(-5,-10) \) - \( D(-5,3) \) Is quadrilateral \( A B C D \) a rhombus, and why? Choose 1 answer: (A) Yes, because \( A B=B C=C D=A D \). (B) Yes, because \( \overline{A B} \| \overline{C D} \), and \( \overline{B C} \| \overline{A D} \). (C) No, because \( \overline{A B} \) is longer than \( \overline{B C} \). (D) No, because \( \overline{A B} \) is not parallel to \( \overline{B C} \).

To determine if quadrilateral \( A B C D \) is a rhombus, we need to check the lengths of its sides as well as the properties of the diagonals. Calculating the lengths of each side: - \( \overline{AB} = \sqrt{(6-6)^2 + (8 - (-5))^2} = \sqrt{0 + 13^2} = 13 \) - \( \overline{BC} = \sqrt{(-5 - 6)^2 + (-10 - (-5))^2} = \sqrt{(-11)^2 + (-5)^2} = \sqrt{121 + 25} = \sqrt{146} \) - \( \overline{CD} = \sqrt{(-5 - (-5))^2 + (3 - (-10))^2} = \sqrt{0 + 13^2} = 13 \) - \( \overline{DA} = \sqrt{(6 - (-5))^2 + (8 - 3)^2} = \sqrt{11^2 + 5^2} = \sqrt{121 + 25} = \sqrt{146} \) We see that \( \overline{AB} = \overline{CD} = 13 \) and \( \overline{BC} = \overline{DA} = \sqrt{146} \). For a rhombus, all sides must be equal. Since we have two pairs of equal lengths but they are not equal to each other, it follows that quadrilateral \( A B C D \) is not a rhombus. The correct answer is: (C) No, because \( \overline{A B} \) is longer than \( \overline{B C} \).