Given the rectangle ABCD on the coordinate plane. The following of the vertices of the rectangle: A = (0,0) B = (4,4) C = (7,-1) D = (3,3)
The length of side AB in simplest radical form is
The length of side BC in simplest radical form is
The area of the rectangle ABCD in simplest radical form is
Answer choices
√32, 4√2, 2√4, √18, 3√2, 9√2, 24, 14√2
First, let's calculate the lengths of the sides AB and BC using the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For side AB:
\[ A = (0,0), \quad B = (4,4) \]
\[ \text{Distance AB} = \sqrt{(4-0)^2 + (4-0)^2} \]
\[ \text{Distance AB} = \sqrt{16+16} \]
\[ \text{Distance AB} = \sqrt{32} \]
\[ \text{Distance AB} = \sqrt{32} = 4 \sqrt{2} \] (simplified form)
Now, for side BC:
\[ B = (4,4), \quad C = (7,-1) \]
\[ \text{Distance BC} = \sqrt{(7-4)^2 + (-1-4)^2} \]
\[ \text{Distance BC} = \sqrt{3^2 + (-5)^2} \]
\[ \text{Distance BC} = \sqrt{9+25} \]
\[ \text{Distance BC} = \sqrt{34} \]
\[ \text{Distance BC} \] is not on your answer choices, so it is possible that the given vertices do not form a rectangle or there has been a mistake in the given vertices.
If we ignore that, for a moment, and try to estimate the area using AB and BC as adjacent sides, we would get:
\[ \text{Area} = \text{Length AB} \times \text{Length BC} = 4 \sqrt{2} \times \sqrt{34} \]
\[ \text{Area} = 4 \sqrt{68} \]
\[ \text{Area} = 4 \times 2 \sqrt{17} \]
\[ \text{Area} = 8 \sqrt{17} \]
However, this is not correct since \(\sqrt{34}\) doesn't match any of the given answer choices and is actually not the distance of BC.
There seems to be an issue with the vertices provided. With vertices A(0,0), B(4,4), C(7,-1), and D(3,3), we cannot form a rectangle. If we assume that one of the points is incorrect, let's say C should be (x, -1) such that it would make CB perpendicular to AB and DC parallel to AB:
AB and CD should be parallel to each other, and BC and AD should be parallel to each other for ABCD to be a rectangle. However, with the given coordinates, AB is at a 45-degree angle to the x-axis, and BC is not perpendicular to AB (as required in a rectangle). Similarly, DC is not parallel to AB, making it not follow the rules of a rectangle.
In conclusion, we cannot form a correct rectangle ABCD with the given vertices, and we cannot correctly calculate the lengths or area of a rectangle because the given vertices do not form a rectangle. Please check the vertices again.