Question

Complete the coordinate proof of the theorem. Given: A B C D is a parallelogram. Prove: The diagonals of A B C D bisect each other. Art: A parallelogram is graphed on a coordinate plane. The horizontal x-axis and vertical y-axis are solid. The vertex labeled as A lies on begin ordered pair 0 comma 0 end ordered pair. The vertex labeled as B lies on begin ordered pair a comma 0 end ordered pair. The vertex labeled as D lies on begin ordered pair c comma b end ordered pair. The vertex C is unlabeled. Diagonals A C and B D are drawn by a dotted lines. Enter your answers in the boxes. The coordinates of parallelogram ABCD are A(0, 0) , B(a, 0) , C( , ), and D(c, b). The coordinates of the midpoint of AC¯¯¯¯¯ are ( , b2 ). The coordinates of the midpoint of BD¯¯¯¯¯ are ( a+c2 , ). The midpoints of the diagonals have the same coordinates. Therefore, AC¯¯¯¯¯ and BD¯¯¯¯¯ bisect each other

Answer 1

To prove that the diagonals of parallelogram ABCD bisect each other using coordinate geometry, one determines vertex C's coordinates and then shows the midpoints of diagonals AC and BD are the same.

Step-by-step explanation:

To complete the coordinate proof that the diagonals of parallelogram ABCD bisect each other, we need to determine the coordinates of vertex C and then show that the midpoints of diagonals AC and BD are the same.

To begin, since AD is parallel to BC and AB is parallel to DC, the coordinates of C must be the sum of the vectors AB and AD, which gives us C(a+c, b). By calculating the midpoints of AC and BD, we find that both midpoints are ((a+c)/2, b/2).

Thus, we have shown that the midpoints for both diagonals are the same, which concludes the proof that the diagonals of parallelogram ABCD bisect each other.