To prove quadrilateral [tex]\(KITE\)[/tex] with vertices [tex]\(K (0, -2), I (1, 2), T (7, 5),\)[/tex] and [tex]\(E (4, -1)\)[/tex] is a kite, we need to compute the distances between consecutive vertices using the distance formula.
The distance formula is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
1. Calculate [tex]\(KI\)[/tex]:
[tex]\[
KI = \sqrt{(1 - 0)^2 + (2 - (-2))^2} = \sqrt{1^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17} \approx 4.123105625617661
\][/tex]
2. Calculate [tex]\(KE\)[/tex]:
[tex]\[
KE = \sqrt{(4 - 0)^2 + (-1 - (-2))^2} = \sqrt{4^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17} \approx 4.123105625617661
\][/tex]
3. Calculate [tex]\(IT\)[/tex]:
[tex]\[
IT = \sqrt{(7 - 1)^2 + (5 - 2)^2} = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} \approx 6.708203932499369
\][/tex]
4. Calculate [tex]\(TE\)[/tex]:
[tex]\[
TE = \sqrt{(7 - 4)^2 + (5 - (-1))^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} \approx 6.708203932499369
\][/tex]
Given these calculations, it is evident that the lengths [tex]\(KI\)[/tex] and [tex]\(KE\)[/tex] are equal, and the lengths [tex]\(IT\)[/tex] and [tex]\(TE\)[/tex] are also equal. This demonstrates that quadrilateral [tex]\(KITE\)[/tex] has two pairs of adjacent sides of equal length, which is a defining characteristic of a kite.
Now, let's complete the sentences with the correct answers:
- [tex]\(KI = \sqrt{(2 - -2)^2 + (1 - 0)^2} = \sqrt{17}\)[/tex]
- [tex]\(KE = \sqrt{17}\)[/tex]
- [tex]\(IT = \sqrt{45}\)[/tex]
- [tex]\(TE = \sqrt{45}\)[/tex]
Therefore, [tex]\(KITE\)[/tex] is a kite because it has two pairs of consecutive sides with equal lengths.
