Let's complete the steps in the proof that show quadrilateral [tex]\( KITE \)[/tex] is a kite.
First, we need to find the distances between the points [tex]\( K, I, T, \)[/tex] and [tex]\( E \)[/tex] using the distance formula:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
1. Calculate [tex]\( KI \)[/tex]:
Coordinates for [tex]\( K \)[/tex] are [tex]\( (0, -2) \)[/tex] and for [tex]\( I \)[/tex] are [tex]\( (1, 2) \)[/tex].
Applying the distance formula:
[tex]\[ KI = \sqrt{(1 - 0)^2 + (2 - (-2))^2} = \sqrt{1^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17} \approx 4.123 \][/tex]
2. Calculate [tex]\( KE \)[/tex]:
Coordinates for [tex]\( K \)[/tex] are [tex]\( (0, -2) \)[/tex] and for [tex]\( E \)[/tex] are [tex]\( (4, -1) \)[/tex].
Applying the distance formula:
[tex]\[ KE = \sqrt{(4 - 0)^2 + (-1 - (-2))^2} = \sqrt{4^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17} \approx 4.123 \][/tex]
3. Calculate [tex]\( IT \)[/tex]:
Coordinates for [tex]\( I \)[/tex] are [tex]\( (1, 2) \)[/tex] and for [tex]\( T \)[/tex] are [tex]\( (7, 5) \)[/tex].
Applying the distance formula:
[tex]\[ IT = \sqrt{(7 - 1)^2 + (5 - 2)^2} = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} \approx 6.708 \][/tex]
4. Calculate [tex]\( TE \)[/tex]:
Coordinates for [tex]\( T \)[/tex] are [tex]\( (7, 5) \)[/tex] and for [tex]\( E \)[/tex] are [tex]\( (4, -1) \)[/tex].
Applying the distance formula:
[tex]\[ TE = \sqrt{(4 - 7)^2 + (-1 - 5)^2} = \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} \approx 6.708 \][/tex]
Therefore, filling in the blanks, we have:
Using the distance formula,
[tex]\[ KI = \sqrt{17} \][/tex]
[tex]\[ KE = \sqrt{17} \][/tex]
[tex]\[ IT = \sqrt{45} \][/tex]
[tex]\[ TE = \sqrt{45} \][/tex]
To complete the proof:
Therefore, KITE is a kite because [tex]\( KI = KE \)[/tex] and [tex]\( IT = TE \)[/tex], indicating that each pair of consecutive sides is equal, which satisfies the properties of a kite.
