Given:
The four points (0,-1), (6,7), (-2,3), and (8,3) are the vertices of a rectangle.

To find:
The area of the rectangle.

Solution:

To determine if the given points form a rectangle, we need to check if the lengths of the sides and the diagonals satisfy the properties of a rectangle.

Step 1: Calculate the lengths of the sides.
Let's calculate the lengths of the sides using the distance formula:

Side AB:
Length AB = √((x2 - x1)^2 + (y2 - y1)^2)
= √((6 - 0)^2 + (7 - (-1))^2)
= √(6^2 + 8^2)
= √(36 + 64)
= √100
= 10

Side BC:
Length BC = √((x2 - x1)^2 + (y2 - y1)^2)
= √((-2 - 6)^2 + (3 - 7)^2)
= √((-8)^2 + (-4)^2)
= √(64 + 16)
= √80
= 4√5

Side CD:
Length CD = √((x2 - x1)^2 + (y2 - y1)^2)
= √((8 - (-2))^2 + (3 - 3)^2)
= √((8 + 2)^2 + (0)^2)
= √(10^2 + 0^2)
= √100
= 10

Side DA:
Length DA = √((x2 - x1)^2 + (y2 - y1)^2)
= √((8 - 0)^2 + (3 - (-1))^2)
= √((8)^2 + (4)^2)
= √(64 + 16)
= √80
= 4√5

Step 2: Check if the opposite sides are equal.
We can see that both sides AB and CD have a length of 10, and both sides BC and DA have a length of 4√5. This satisfies the property of a rectangle, where opposite sides are equal.

Step 3: Check if the diagonals are equal.
The length of the diagonal AC can be calculated using the distance formula:
Length AC = √((x2 - x1)^2 + (y2 - y1)^2)
= √((8 - 0)^2 + (3 - (-1))^2)
= √((8)^2 + (4)^2)
= √(64 + 16)
= √80
= 4√5

Similarly, the length of the diagonal BD can be calculated:
Length BD = √((x2 - x1)^2 + (y2 - y1)^2)
= √((-2 - 6)^2 + (3 - 7)^2)
= √((-8)^2 + (-4)^2)
=