Question

Given that the following coordinates are the vertices of a rectangle, prove that this truly is a rectangle by showing the slopes of the sides that meet are perpendicular:

(-1, 1), (2, 0), (3, 3), and (0, 4).
A) The slopes are perpendicular because the product of their slopes is -1.
B) The slopes are perpendicular because they are equal.
C) The slopes are not perpendicular.
D) The slopes are perpendicular because the sum of their slopes is -1.

Answer 1

To prove that the given coordinates form a rectangle, we need to show that the slopes of the sides that meet are perpendicular. The slopes are not perpendicular because the product of the slopes is not -1.

Step-by-step explanation:

To prove that the given coordinates (-1, 1), (2, 0), (3, 3), and (0, 4) form a rectangle, we need to show that the slopes of the sides that meet are perpendicular. The slopes of the sides can be determined by finding the difference in the y-coordinates divided by the difference in the x-coordinates. Let's calculate the slopes:

1. Slope of side 1: (-1, 1) and (2, 0)
   (0 - 1) / (2 - (-1)) = -1/3
2. Slope of side 2: (2, 0) and (3, 3)
   (3 - 0) / (3 - 2) = 3/1 = 3
3. Slope of side 3: (3, 3) and (0, 4)
   (4 - 3) / (0 - 3) = -1/3
4. Slope of side 4: (0, 4) and (-1, 1)
   (1 - 4) / (-1 - 0) = 3/1 = 3

The slopes are not equal, so option B is incorrect. The product of the slopes of the sides that meet is not -1, so option A is incorrect. However, the slopes of side 1 and side 3 are perpendicular because their product is -1. Therefore, the answer is option C: The slopes are not perpendicular.