Final answer:
By calculating the slopes of the sides and showing they are perpendicular to each other, and then doing the same for the diagonals, we have proved the quadrilateral is a rectangle, and its diagonals are also perpendicular to each other.
Step-by-step explanation:
To prove whether a quadrilateral with vertices (-1, 1), (2, 0), (3, 3), and (0, 4) is a rectangle, we need to show that the slopes of the sides that meet are perpendicular. The slope is determined by the formula (y2 - y1)/(x2 - x1), which represents the 'rise over run'. Let's calculate the slopes of each side:
- Side 1 (between (-1, 1) and (2, 0)): slope = (0 - 1) / (2 - (-1)) = -1 / 3
- Side 2 (between (2, 0) and (3, 3)): slope = (3 - 0) / (3 - 2) = 3
- Side 3 (between (3, 3) and (0, 4)): slope = (4 - 3) / (0 - 3) = 1 / -3
- Side 4 (between (0, 4) and (-1, 1)): slope = (1 - 4) / (-1 - 0) = -3
The slopes of Side 1 and Side 4, as well as Side 2 and Side 3, are negative reciprocals of each other, which indicates they are perpendicular. Hence, these sides meet at right angles, and the quadrilateral is a rectangle.
Now, let's find the slopes of the diagonals:
- Diagonal 1 (between (-1, 1) and (3, 3)): slope = (3 - 1) / (3 - (-1)) = 2 / 4 = 1/2
- Diagonal 2 (between (2, 0) and (0, 4)): slope = (4 - 0) / (0 - 2) = 4 / -2 = -2
The slopes of the diagonals are also negative reciprocals, which means that the diagonals are perpendicular to each other. This is another characteristic of a rectangle.
Considering the information provided, the answer is 'A. The diagonals of the rectangle are perpendicular'.