To confirm that quadrilateral KITE with vertices [tex]\( K(0, -2) \)[/tex], [tex]\( I(1, 2) \)[/tex], [tex]\( T(7, 5) \)[/tex], and [tex]\( E(4, -1) \)[/tex] is a kite, we will use the distance formula to calculate the lengths of its sides and then analyze the properties of these sides:
1. Calculating [tex]\( KI \)[/tex]:
[tex]\[
KI = \sqrt{(2 - (-2))^2 + (1 - 0)^2} = \sqrt{(4)^2 + (1)^2} = \sqrt{16 + 1} = \sqrt{17} \approx 4.123
\][/tex]
2. Calculating [tex]\( KE \)[/tex]:
[tex]\[
KE = \sqrt{(-1 - (-2))^2 + (4 - 0)^2} = \sqrt{(1)^2 + (4)^2} = \sqrt{1 + 16} = \sqrt{17} \approx 4.123
\][/tex]
3. Calculating [tex]\( IT \)[/tex]:
[tex]\[
IT = \sqrt{(5 - 2)^2 + (7 - 1)^2} = \sqrt{(3)^2 + (6)^2} = \sqrt{9 + 36} = \sqrt{45} \approx 6.708
\][/tex]
4. Calculating [tex]\( TE \)[/tex]:
[tex]\[
TE = \sqrt{(5 - (-1))^2 + (7 - 4)^2} = \sqrt{(6)^2 + (3)^2} = \sqrt{36 + 9} = \sqrt{45} \approx 6.708
\][/tex]
Since we have:
- [tex]\( KI \approx 4.123 \)[/tex]
- [tex]\( KE \approx 4.123 \)[/tex]
- [tex]\( IT \approx 6.708 \)[/tex]
- [tex]\( TE \approx 6.708 \)[/tex]
We notice that [tex]\( KI = KE \)[/tex] and [tex]\( IT = TE \)[/tex].
Therefore, quadrilateral [tex]\( \operatorname{KITE} \)[/tex] is a kite because it has two pairs of adjacent sides that are equal.
