To show that quadrilateral KITE is a kite using the given vertices \( K(0, -2) \), \( I(1, 2) \), \( T(7, 5) \), and \( E(4, -1) \), we follow these steps:
1. Calculate the lengths of the sides:
- Length \(KI\):
The distance between \(K(0, -2)\) and \(I(1, 2)\) is given by:
[tex]\[
KI = \sqrt{(1 - 0)^2 + (2 - (-2))^2} = \sqrt{1^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17} \approx 4.12
\][/tex]
- Length \(IT\):
The distance between \(I(1, 2)\) and \(T(7, 5)\) is given by:
[tex]\[
IT = \sqrt{(7 - 1)^2 + (5 - 2)^2} = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} \approx 6.71
\][/tex]
- Length \(TE\):
The distance between \(T(7, 5)\) and \(E(4, -1)\) is given by:
[tex]\[
TE = \sqrt{(4 - 7)^2 + (-1 - 5)^2} = \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} \approx 6.71
\][/tex]
- Length \(EK\):
The distance between \(E(4, -1)\) and \(K(0, -2)\) is given by:
[tex]\[
EK = \sqrt{(0 - 4)^2 + (-2 - (-1))^2} = \sqrt{(-4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \approx 4.12
\][/tex]
2. Check if KITE is a kite:
A kite is defined as a quadrilateral with two pairs of adjacent sides that are equal in length. In this case:
- \(KI \approx 4.12\)
- \(IT \approx 6.71\)
- \(TE \approx 6.71\)
- \(EK \approx 4.12\)
We observe that:
- \(KI = EK \approx 4.12\)
- \(IT = TE \approx 6.71\)
Since \(KITE\) has two pairs of adjacent sides that are equal, quadrilateral KITE is indeed a kite.
In summary:
- \(KI = \sqrt{17}\)
- \(IT = \sqrt{45}\)
- \(TE = \sqrt{45}\)
- \(EK = \sqrt{17}\)
Therefore, KITE is a kite because it has two pairs of adjacent sides that are equal in length.
