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.
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To show that the diagonals of quadrilateral \( A B C D \) bisect each other, you would want to use the concept of midpoints. Specifically, option C is the way to go! If the midpoints of both diagonals \( \overline{A C} \) and \( \overline{B D} \) are the same, it means they intersect at that midpoint, confirming that the diagonals bisect each other. In coordinate geometry, calculating the midpoint is straightforward! You simply average the x-coordinates and the y-coordinates of the endpoints. So for diagonal \( \overline{A C} \), the midpoint can be found using \( M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \). If you find that the midpoints of both diagonals are the same, voilà, you’ve proven they bisect each other!
