Question

"Prove that quadrilateral MNOP is a parallelogram by showing that its opposite sides are parallel. The coordinates of the vertices of MNOP are M(-2, -4), N(1, 2), O(2, 10), and P(-1, 4). Determine the slopes of the line segments MN, NO, OP, and PM. Show that opposite sides have equal slopes, which would confirm that MNOP is a parallelogram."

Answer 1

To prove that quadrilateral MNOP is a parallelogram, we need to show that its opposite sides are parallel. By finding the slopes of the line segments MN, NO, OP, and PM and showing that opposite sides have equal slopes, we can confirm that MNOP is a parallelogram.

Step-by-step explanation:

To prove that quadrilateral MNOP is a parallelogram, we need to show that its opposite sides are parallel. First, we find the slopes of the line segments MN, NO, OP, and PM.

For segment MN, the slope is (2 - (-4))/(1 - (-2)) = 6/3 = 2.

For segment NO, the slope is (10 - 2)/(2 - 1) = 8/1 = 8.

For segment OP, the slope is (4 - 10)/(-1 - 2) = -6/-3 = 2.

For segment PM, the slope is (-4 - 4)/(-2 - (-1)) = -8/-1 = 8.

As we can see, the opposite sides MN and OP have the same slope of 2, and the opposite sides NO and PM have the same slope of 8.

Therefore, the opposite sides of MNOP are parallel, confirming that MNOP is a parallelogram.