To determine if the quadrilateral KITE is a kite given the vertices [tex]\( K(0, -2) \)[/tex], [tex]\( I(1, 2) \)[/tex], [tex]\( T(7, 5) \)[/tex], and [tex]\( E(4, -1) \)[/tex], we need to use the distance formula:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
1. Calculate [tex]\( KI \)[/tex]:
[tex]\[
KI = \sqrt{(1 - 0)^2 + (2 - (-2))^2} = \sqrt{1^2 + (4)^2} = \sqrt{1 + 16} = \sqrt{17}
\][/tex]
2. Calculate [tex]\( KE \)[/tex]:
[tex]\[
KE = \sqrt{(4 - 0)^2 + (-1 - (-2))^2} = \sqrt{4^2 + (1)^2} = \sqrt{16 + 1} = \sqrt{17}
\][/tex]
3. Calculate [tex]\( IT \)[/tex]:
[tex]\[
IT = \sqrt{(7 - 1)^2 + (5 - 2)^2} = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}
\][/tex]
4. Calculate [tex]\( TE \)[/tex]:
[tex]\[
TE = \sqrt{(7 - 4)^2 + (5 - (-1))^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}
\][/tex]
Therefore:
- [tex]\( KI = \sqrt{17} \)[/tex]
- [tex]\( KE = \sqrt{17} \)[/tex]
- [tex]\( IT = 3\sqrt{5} \)[/tex]
- [tex]\( TE = 3\sqrt{5} \)[/tex]
These calculations show that [tex]\( KI = KE \)[/tex] and [tex]\( IT = TE \)[/tex].
Therefore, KITE is a kite because it has two pairs of adjacent sides that are equal in length: [tex]\( KI = KE \)[/tex] and [tex]\( IT = TE \)[/tex].
