To find the perimeter of the kite, we need to calculate the lengths of its sides and then sum these lengths.
First, label the vertices of the kite as follows:
- [tex]\( A = (2, 4) \)[/tex]
- [tex]\( B = (5, 4) \)[/tex]
- [tex]\( C = (5, 1) \)[/tex]
- [tex]\( D = (0, -1) \)[/tex]
Next, we calculate the distances between consecutive vertices using the distance formula:
[tex]\[ \text{Distance between } (x_1, y_1) \text{ and } (x_2, y_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
1. Calculate the distance [tex]\( AB \)[/tex]:
[tex]\[ AB = \sqrt{(5 - 2)^2 + (4 - 4)^2} = \sqrt{3^2 + 0^2} = \sqrt{9} = 3.0 \][/tex]
2. Calculate the distance [tex]\( BC \)[/tex]:
[tex]\[ BC = \sqrt{(5 - 5)^2 + (1 - 4)^2} = \sqrt{0^2 + (-3)^2} = \sqrt{9} = 3.0 \][/tex]
3. Calculate the distance [tex]\( CD \)[/tex]:
[tex]\[ CD = \sqrt{(5 - 0)^2 + (1 - (-1))^2} = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29} \approx 5.4 \][/tex]
4. Calculate the distance [tex]\( DA \)[/tex]:
[tex]\[ DA = \sqrt{(0 - 2)^2 + (-1 - 4)^2} = \sqrt{(-2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.4 \][/tex]
To find the perimeter of the kite, we sum these distances:
[tex]\[ \text{Perimeter} = AB + BC + CD + DA \approx 3.0 + 3.0 + 5.4 + 5.4 = 16.8 \][/tex]
Thus, the approximate perimeter of the kite, rounded to the nearest tenth, is:
[tex]\[ \boxed{16.8} \][/tex]
