Question

A polygon has vertices whose coordinates are A(1, 4), B(4, -1), C(-1, -4), and D(-4, 1). Use the midpoint formula to determine whether or not the diagonals of polygon ABCD bisect each other. In your final answer, include all of your calculations.

a. The diagonals bisect each other.
b. The diagonals do not bisect each other.

Answer 1

Using the midpoint formula, it was determined that both diagonals (AC and BD) of the polygon bisect each other at the origin (0, 0), hence, the diagonals do bisect each other.

Step-by-step explanation:

To determine whether the diagonals of the polygon with vertices A(1, 4), B(4, -1), C(-1, -4), and D(-4, 1) bisect each other, we need to find the midpoints of diagonals AC and BD using the midpoint formula, which is given by M = ((x1 + x2)/2, (y1 + y2)/2) for a diagonal with endpoints (x1, y1) and (x2, y2).

For diagonal AC, the midpoint M1 is calculated as follows:

M1 = ((1 - 1)/2, (4 - 4)/2) = (0/2, 0/2) = (0, 0)

For diagonal BD, the midpoint M2 is calculated as follows:

M2 = ((4 + -4)/2, (-1 + 1)/2) = (0/2, 0/2) = (0, 0)

Since M1 and M2 have the same coordinates, it indicates that both diagonals AC and BD bisect each other at the origin, i.e., point (0, 0). Thus, the diagonals do bisect each other.