Parallelogram \( A B C D \) has the following vertices: - \( A(0,8) \) - \( B(8,4) \) - \( C(2,-8) \) I \( (-6,-4) \) Is parallelogram \( A B C D \) a rectangle, and why? Choose 1 answer: (A) Yes, because \( A B=A D \) and \( B C=C D \), and \( A B C D \) is a parallelogram. (B) Yes, because \( \overline{B C} \) is perpendicular to \( \overline{A B} \), and \( A B C D \) is a parallelogram. (C) No, because \( \overline{A B} \) is shorter than \( \overline{A D} \). (D) No, because \( \overline{B C} \) is not perpendicular to \( \overline{A B} \).

We first compute the vectors representing two adjacent sides. 1. Compute vector \(\overrightarrow{AB}\): \[ \overrightarrow{AB} = B - A = (8-0,\; 4-8) = (8,\; -4) \] 2. Compute vector \(\overrightarrow{BC}\): \[ \overrightarrow{BC} = C - B = (2-8,\; -8-4) = (-6,\; -12) \] 3. Check for perpendicularity by computing the dot product of \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\): \[ \overrightarrow{AB} \cdot \overrightarrow{BC} = (8)(-6) + (-4)(-12) = -48 + 48 = 0 \] Since the dot product is \(0\), the vectors \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\) are perpendicular. Because a parallelogram with one right angle is a rectangle, parallelogram \(ABCD\) is a rectangle. Thus, the correct answer is: (B) Yes, because \(\overline{BC}\) is perpendicular to \(\overline{AB}\), and \(ABCD\) is a parallelogram.