Question

Given the rectangle ABCD on the coordinate plane.
a=(0,0) b=(4,-4) c=(7,-1) d=(3,3)
(3 points)
The length of side AB in simplest radical form is Response area.

The length of side BC in simplest radical form is Response area.

The area of rectangle ABCD in simplest radical form is Response area.

Answer 1

To find the lengths of the sides AB and BC, as well as the area of rectangle ABCD, we can use the distance formula and the properties of the rectangle.

1. **Length of side AB**:

Points A and B are given as:
- A = (0, 0)
- B = (4, -4)

Using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
we find the length of AB:
\[
AB = \sqrt{(4 - 0)^2 + (-4 - 0)^2} = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\]

2. **Length of side BC**:

Points B and C are given as:
- B = (4, -4)
- C = (7, -1)

Using the distance formula again:
\[
BC = \sqrt{(7 - 4)^2 + (-1 - (-4))^2} = \sqrt{(3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]

3. **Area of rectangle ABCD**:

The area of a rectangle is given by the product of the lengths of two adjacent sides. Therefore, the area A is:
\[
A = AB \times BC = (4\sqrt{2}) \times (3\sqrt{2}) = 12 \times 2 = 24
\]

Summarizing the findings:
- The length of side AB in simplest radical form is \(4\sqrt{2}\).
- The length of side BC in simplest radical form is \(3\sqrt{2}\).
- The area of rectangle ABCD in simplest radical form is \(24\).