Let's find the area and perimeter of the quadrilateral [tex]\(ABCD\)[/tex] with the given vertices:
1. Calculate the lengths of each side using the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
- Length of [tex]\(AB\)[/tex]:
[tex]\[
AB = \sqrt{(2 - (-1))^2 + (3 - (-1))^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.0
\][/tex]
- Length of [tex]\(BC\)[/tex]:
[tex]\[
BC = \sqrt{(5 - 2)^2 + (3 - 3)^2} = \sqrt{3^2 + 0^2} = \sqrt{9} = 3.0
\][/tex]
- Length of [tex]\(CD\)[/tex]:
[tex]\[
CD = \sqrt{(8 - 5)^2 + (-1 - 3)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5.0
\][/tex]
- Length of [tex]\(DA\)[/tex]:
[tex]\[
DA = \sqrt{(8 - (-1))^2 + (-1 - (-1))^2} = \sqrt{9^2 + 0^2} = \sqrt{81} = 9.0
\][/tex]
2. Calculate the perimeter by summing the lengths of all sides:
[tex]\[
\text{Perimeter} = AB + BC + CD + DA = 5.0 + 3.0 + 5.0 + 9.0 = 22.0 \, \text{units}
\][/tex]
3. Calculate the area using the Shoelace formula:
The Shoelace formula for the area of a quadrilateral given its vertices [tex]\((x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4)\)[/tex] is:
[tex]\[
\text{Area} = \frac{1}{2} \left| x_1 y_2 + x_2 y_3 + x_3 y_4 + x_4 y_1 - (y_1 x_2 + y_2 x_3 + y_3 x_4 + y_4 x_1) \right|
\][/tex]
Plugging in the coordinates:
[tex]\[
\text{Area} = \frac{1}{2} \left| (-1 \cdot 3) + (2 \cdot 3) + (5 \cdot -1) + (8 \cdot -1) - ((-1 \cdot 2) + (3 \cdot 5) + (3 \cdot 8) + (-1 \cdot -1)) \right|
\][/tex]
[tex]\[
= \frac{1}{2} \left| (-3) + 6 + (-5) + (-8) - ( -2 + 15 + 24 + 1) \right|
\][/tex]
[tex]\[
= \frac{1}{2} \left| -10 - 38 \right|
\][/tex]
[tex]\[
= \frac{1}{2} \cdot 48 = 24.0 \, \text{square units}
\][/tex]
Therefore, the area of the quadrilateral [tex]\(ABCD\)[/tex] is [tex]\(24.0 \, u^2\)[/tex] and the perimeter is [tex]\(22.0 \, u\)[/tex].
