To determine whether quadrilateral KITE is a kite, we will compute the lengths of its sides using the distance formula. A kite is a quadrilateral with two pairs of adjacent sides that are equal in length.
Given vertices:
- [tex]\( K (0, -2) \)[/tex]
- [tex]\( I (1, 2) \)[/tex]
- [tex]\( T (7, 5) \)[/tex]
- [tex]\( E (4, -1) \)[/tex]
1. Calculate [tex]\( K I \)[/tex]:
[tex]\[
K I = \sqrt{(2 - (-2))^2 + (1 - 0)^2} = \sqrt{(2 + 2)^2 + (1 - 0)^2} = \sqrt{4^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17}
\][/tex]
2. Calculate [tex]\( K E \)[/tex]:
[tex]\[
K E = \sqrt{(-1 - (-2))^2 + (4 - 0)^2} = \sqrt{(-1 + 2)^2 + 4^2} = \sqrt{1^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17}
\][/tex]
3. Calculate [tex]\( I T \)[/tex]:
[tex]\[
I T = \sqrt{(5 - 2)^2 + (7 - 1)^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45}
\][/tex]
Simplifying the square root gives us:
[tex]\[
I T = \sqrt{45} = 3\sqrt{5} \approx 6.708
\][/tex]
4. Calculate [tex]\( T E \)[/tex]:
[tex]\[
T E = \sqrt{(5 - (-1))^2 + (7 - 4)^2} = \sqrt{(5 + 1)^2 + (3)^2} = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45}
\][/tex]
Simplifying the square root gives us:
[tex]\[
T E = \sqrt{45} = 3\sqrt{5} \approx 6.708
\][/tex]
Comparing these calculated distances:
- [tex]\( K I = \sqrt{17} \approx 4.123 \)[/tex]
- [tex]\( K E = \sqrt{17} \approx 4.123 \)[/tex]
- [tex]\( I T = \sqrt{45} = 3\sqrt{5} \approx 6.708 \)[/tex]
- [tex]\( T E = \sqrt{45} = 3\sqrt{5} \approx 6.708 \)[/tex]
Since [tex]\( KI \)[/tex] equals [tex]\( KE \)[/tex] and [tex]\( IT \)[/tex] equals [tex]\( TE \)[/tex], we have two pairs of adjacent sides that are equal. Therefore, quadrilateral KITE is a kite.
[tex]\[
\text{Using the distance formula, } KI=\sqrt{17}, KE=\sqrt{17}, IT=\sqrt{45}, \text{ and } TE= \sqrt{45}
\][/tex]
Therefore, KITE is a kite because [tex]\( KI = KE \)[/tex] and [tex]\( IT = TE \)[/tex].
