Question

Hiroshi is writing a coordinate proof to show that the diagonals of a parallelogram bisect each other. He begins by assigning coordinates to the vertices of a parallelogram as shown. Which sentence describes what Hiroshi should do to show that the diagonals of the parallelogram bisect each other?

1) Show that the midpoint of segment MP is (a/2, b/2, c/2) and the midpoint of segment QN is (a/2, b/2, c/2).
2) Show that MN = QP = a.
3) Show that the slope of segment QM is c/b and the slope of segment PN is c/b.
4) Show that the length of segment QN is greater than the length of segment MP.

Answer 1

To show that the diagonals of a parallelogram bisect each other, Hiroshi should calculate and demonstrate that the midpoints of both diagonals coincide, using methods based on the Pythagorean theorem and the properties of parallelograms.

Step-by-step explanation:

The student is asked to determine what Hiroshi should do to show that the diagonals of a parallelogram bisect each other. Hiroshi should first assign coordinates to the vertices of a parallelogram following certain guidelines. Then, to show that the diagonals of the parallelogram bisect each other, Hiroshi should, ideally, calculate the midpoints of both diagonals and demonstrate that these midpoints coincide. Specifically, Hiroshi should show that the midpoint of segment MP is (a/2, b/2, c/2) and the midpoint of segment QN is also (a/2, b/2, c/2). This would confirm that the diagonals of the parallelogram bisect each other at the same point.

However, because the provided information seems unrelated to the actual process of proving that the diagonals of a parallelogram bisect each other using coordinate geometry, Hiroshi must rely on general methods. Using the Pythagorean theorem and understanding of the properties of parallelograms, Hiroshi would show that the midpoint of one diagonal is equal to the midpoint of the other diagonal, proving they bisect each other.