Rectangle \( A B C D \) has the following vertices: - \( A(-1,9) \) D(9,4) - \( C(4,-6) \) Is rectangle \( A B C D \) a square, and why? Choose 1 answer: (A) Yes, because \( A B=B C=C D=A D \), and \( A B C D \) is a rectangle. (B) Yes, because \( \overline{A B} \| \overline{C D} \) and \( \overline{B C} \| \overline{A D} \), and \( A B C D \) is a rectangle. (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 rectangle \( A B C D \) is a square, we need to calculate the lengths of the sides and check if they are equal. First, let's find the coordinates of point \( B \). Since \( A \) and \( C \) are two opposing vertices of the rectangle, \( B \) must share a vertical position with \( A \) and a horizontal position with \( C \). Thus, \( B \) will be at \( (4, 9) \). Now, we can find the lengths of the sides: - Length of \( A B \): \[ A(-1,9) \text{ to } B(4,9) \Rightarrow |4 - (-1)| = 5 \] - Length of \( B C \): \[ B(4,9) \text{ to } C(4,-6) \Rightarrow |9 - (-6)| = 15 \] - Length of \( C D \): \[ C(4,-6) \text{ to } D(9,4) \Rightarrow |9 - 4| + |4 - (-6)| = 5 + 10 = 15 \] - Length of \( D A \): \[ D(9,4) \text{ to } A(-1,9) \Rightarrow |9 - (-1)| + |4 - 9| = 10 + 5 = 15 \] Since the four sides are not equal (\( A B \) is shorter than \( B C \)), this indicates that rectangle \( A B C D \) is not a square. Therefore, the answer is: (C) No, because \( \overline{A B} \) is longer than \( \overline{B C} \).