To determine the approximate perimeter of a kite with vertices at [tex]\((2, 4)\)[/tex], [tex]\((5, 4)\)[/tex], [tex]\((5, 1)\)[/tex], and [tex]\((0, -1)\)[/tex], we need to calculate the lengths of its four sides and then sum these lengths. Here are the detailed steps:
1. Calculate the distance between the vertices [tex]\((2, 4)\)[/tex] and [tex]\((5, 4)\)[/tex]:
These points share the same y-coordinate, so the distance is simply the difference in x-coordinates:
[tex]\[
\text{Distance} = \sqrt{(5 - 2)^2 + (4 - 4)^2} = \sqrt{3^2 + 0^2} = \sqrt{9} = 3.0
\][/tex]
2. Calculate the distance between the vertices [tex]\((5, 4)\)[/tex] and [tex]\((5, 1)\)[/tex]:
These points share the same x-coordinate, so the distance is simply the difference in y-coordinates:
[tex]\[
\text{Distance} = \sqrt{(5 - 5)^2 + (4 - 1)^2} = \sqrt{0^2 + 3^2} = \sqrt{9} = 3.0
\][/tex]
3. Calculate the distance between the vertices [tex]\((5, 1)\)[/tex] and [tex]\((0, -1)\)[/tex]:
[tex]\[
\text{Distance} = \sqrt{(0 - 5)^2 + (-1 - 1)^2} = \sqrt{(-5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29} \approx 5.385
\][/tex]
4. Calculate the distance between the vertices [tex]\((0, -1)\)[/tex] and [tex]\((2, 4)\)[/tex]:
[tex]\[
\text{Distance} = \sqrt{(2 - 0)^2 + (4 - (-1))^2} = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.385
\][/tex]
Next, we add these distances together to find the perimeter of the kite:
[tex]\[
\text{Perimeter} = 3.0 + 3.0 + 5.385 + 5.385 \approx 16.770
\][/tex]
Rounding this to the nearest tenth:
[tex]\[
\text{Perimeter} \approx 16.8 \text{ units}
\][/tex]
Thus, the approximate perimeter of the kite is [tex]\(16.8\)[/tex] units. Therefore, the correct answer is:
[tex]\[
\boxed{16.8 \text{ units}}
\][/tex]
