Question

A polygon has vertices 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.

Answer 1

After using the midpoint formula to calculate the midpoints of diagonals AC and BD of the polygon, we found that both midpoints are the same, (0, 0). This confirms that the diagonals do indeed 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 the diagonals AC and BD using the midpoint formula. The midpoint formula is given by ((x1 + x2)/2, (y1 + y2)/2), where (x1, y1) and (x2, y2) are the coordinates of the endpoints.

Calculating Midpoint of Diagonal AC:

For points A(1, 4) and C(-1, -4), the midpoint M1 would be:

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

Calculating Midpoint of Diagonal BD:

For points B(4, -1) and D(-4, 1), the midpoint M2 would be:

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

Since both diagonals have the same midpoint, (0, 0), we can conclude that they do bisect each other.