Which of the following would show that the diagonals of quadrilateral \( A B C D \) on the coordinate plane bisect each other? A. Show that slope of \( \overline{A C}= \) the slope of \( \overline{B D} \). B. Show that \( A C=B D \). C. Show that the midpoint of \( \overline{A C} \) is the same as the midpoint of \( \overline{B D} \). Show that (slope of \( \overline{A C} \) )(slope of \( \overline{B D} \) ) \( =-1 \). D.

To determine which option shows that the diagonals of quadrilateral \( ABCD \) bisect each other, we need to understand the concept of bisecting diagonals. When two segments bisect each other, it means that their midpoints are the same. Therefore, we need to find the option that confirms that the midpoints of the diagonals \( AC \) and \( BD \) are equal. Let's analyze the options: A. **Show that slope of \( \overline{A C} = \) the slope of \( \overline{B D} \)**: This only shows that the lines are parallel, not that they bisect each other. B. **Show that \( AC = BD \)**: This indicates that the lengths of the diagonals are equal, but it does not imply that they bisect each other. C. **Show that the midpoint of \( \overline{A C} \) is the same as the midpoint of \( \overline{B D} \). Show that (slope of \( \overline{A C} \) )(slope of \( \overline{B D} \)) \( = -1 \)**: The first part of this option confirms that the midpoints are the same, which is necessary for the diagonals to bisect each other. The second part about the slopes being negative reciprocals indicates that the lines are perpendicular, but it is not necessary for the diagonals to bisect each other. D. **(Not provided)**: We cannot evaluate this option. Based on the analysis, the correct answer is: **C. Show that the midpoint of \( \overline{A C} \) is the same as the midpoint of \( \overline{B D} \).** This confirms that the diagonals bisect each other.