To find the area of the rectangle with given vertices [tex]\((1,7), (5,3), (3,1), \text{ and } (-1,5)\)[/tex], follow these steps:
1. Identify the sides of the rectangle:
A rectangle has two pairs of opposite sides that are equal in length.
2. Use the distance formula to calculate the lengths of the sides:
The distance formula between two points [tex]\((x_1, y_1) \text{ and } (x_2, y_2)\)[/tex] is given by:
[tex]\[
\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
3. Calculate the length between [tex]\((1,7) \text{ and } (5,3)\)[/tex]:
[tex]\[
\text{Length}_1 = \sqrt{(5 - 1)^2 + (3 - 7)^2} = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \approx 5.656854
\][/tex]
4. Calculate the length between [tex]\((3,1) \text{ and } (-1,5)\)[/tex]:
[tex]\[
\text{Length}_2 = \sqrt{(3 - (-1))^2 + (5 - 1)^2} = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \approx 5.656854
\][/tex]
5. The other two sides, by virtue of being a rectangle, will match these lengths exactly. In this case, both sides being equal confirms the sides identified are opposite and therefore a pair.
6. Calculate the area of the rectangle using the formula [tex]\( \text{Area} = \text{side}_1 \times \text{side}_2\)[/tex]:
[tex]\[
\text{Area} = 5.656854 \times 5.656854 = 32 \text{ square units}
\][/tex]
Thus, the area of the rectangle is:
[tex]\[
\boxed{32 \text{ units}^2}
\][/tex]
