To demonstrate 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 calculate the lengths of its sides using the distance formula. The distance formula for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Let's calculate the distances for each side:
1. Distance [tex]\(KI\)[/tex]:
[tex]\[
KI = \sqrt{(1 - 0)^2 + (2 - (-2))^2} = \sqrt{1 + 16} = \sqrt{17}
\][/tex]
2. Distance [tex]\(KE\)[/tex]:
[tex]\[
KE = \sqrt{(4 - 0)^2 + (-1 - (-2))^2} = \sqrt{16 + 1} = \sqrt{17}
\][/tex]
3. Distance [tex]\(IT\)[/tex]:
[tex]\[
IT = \sqrt{(7 - 1)^2 + (5 - 2)^2} = \sqrt{36 + 9} = \sqrt{45} = 6.708203932499369
\][/tex]
4. Distance [tex]\(TE\)[/tex]:
[tex]\[
TE = \sqrt{(4 - 7)^2 + (-1 - 5)^2} = \sqrt{9 + 36} = \sqrt{45} = 6.708203932499369
\][/tex]
From these calculations, we see that:
[tex]\[
KI = KE = \sqrt{17}
\][/tex]
[tex]\[
IT = TE = 6.708203932499369
\][/tex]
Therefore, KITE is a kite because it has two consecutive pairs of adjacent sides with equal lengths.
To finish the sentence with the numerical values:
1. For [tex]\( KE \)[/tex], we have [tex]\( \sqrt{17} \)[/tex]
2. For [tex]\( IT \)[/tex], we have [tex]\( 6.708203932499369 \)[/tex]
3. For [tex]\( TE \)[/tex], we have [tex]\( 6.708203932499369 \)[/tex]
Thus completing the steps in the proof:
Using the distance formula, [tex]\( KI = \sqrt{(2 - (-2))^2 + (1 - 0)^2} = \sqrt{17} \)[/tex], [tex]\( KE = \sqrt{17} \)[/tex], [tex]\( IT = 6.708203932499369 \)[/tex], and [tex]\( TE = 6.708203932499369 \)[/tex].
Therefore, KITE is a kite because two consecutive pairs of adjacent sides have equal lengths [tex]\( \checkmark \)[/tex].
