To prove that quadrilateral ABCD is a rectangle, we can consider the properties of rectangles, which include parallel sides and right angles. Here are the options that could be used to prove ABCD is a rectangle:
A. Use the slope formula to show that \( AB \parallel CD \), \( BC \parallel AD \), and \( AC \perp BD \).
- This option proves that opposite sides are parallel and that the diagonals intersect at right angles, thereby confirming the properties of a rectangle.
C. Use the distance formula to show that \( AB = CD \), \( BC = AD \), and use the midpoint formula to show that the midpoint of \( AC \) and the midpoint of \( BD \) are the same point.
- This option confirms that opposite sides are equal (a requirement for parallelograms), and it confirms that the diagonals bisect each other at the same point, indicating that ABCD is a parallelogram. However, it does not directly prove that the angles are right.
D. Use the distance formula to show that \( AB = CD \), and use the slope formula to show that \( AB \parallel CD \) and \( AB \perp BC \).
- This option shows that one set of opposite sides is equal and that one angle is right, but it does not check the other angle or the other set of sides.
E. Use the slope formula to show that \( AB \parallel CD \), and use the distance formula to show that \( AB = CD \) and \( AC = BD \).
- This could confirm that one set of opposite sides is parallel and that both pairs of opposite sides are equal, but it doesn't explicitly check for right angles.
Based on these options:
- The best choices are A, C, and E.
- A shows both parallel sides and right angles.
- C demonstrates the equality of sides and confirms midpoints.
- E verifies parallelism and equality of lengths but lacks direct right angle proof.
To ensure completeness, Option A is the most thorough in proving that ABCD is a rectangle because it incorporates both conditions for parallelism and perpendicularity, which together imply right angles.