To determine if 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 need to show that it has two pairs of adjacent sides with equal lengths. This can be done using the distance formula:
[tex]\[ \text{Distance between two points } (x_1, y_1) \text{ and } (x_2, y_2) \text{ is given by } \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Let's start by calculating the distances between the given vertices.
1. Calculate [tex]\( KI \)[/tex]:
[tex]\[
KI = \sqrt{(1 - 0)^2 + (2 - (-2))^2}
\][/tex]
[tex]\[
= \sqrt{(1)^2 + (4)^2}
\][/tex]
[tex]\[
= \sqrt{1 + 16}
\][/tex]
[tex]\[
= \sqrt{17}
\][/tex]
[tex]\[
\approx 4.12
\][/tex]
2. Calculate [tex]\( KE \)[/tex]:
[tex]\[
KE = \sqrt{(4 - 0)^2 + (-1 - (-2))^2}
\][/tex]
[tex]\[
= \sqrt{(4)^2 + (1)^2}
\][/tex]
[tex]\[
= \sqrt{16 + 1}
\][/tex]
[tex]\[
= \sqrt{17}
\][/tex]
[tex]\[
\approx 4.12
\][/tex]
3. Calculate [tex]\( IT \)[/tex]:
[tex]\[
IT = \sqrt{(7 - 1)^2 + (5 - 2)^2}
\][/tex]
[tex]\[
= \sqrt{(6)^2 + (3)^2}
\][/tex]
[tex]\[
= \sqrt{36 + 9}
\][/tex]
[tex]\[
= \sqrt{45}
\][/tex]
[tex]\[
\approx 6.71
\][/tex]
4. Calculate [tex]\( TE \)[/tex]:
[tex]\[
TE = \sqrt{(7 - 4)^2 + (5 - (-1))^2}
\][/tex]
[tex]\[
= \sqrt{(3)^2 + (6)^2}
\][/tex]
[tex]\[
= \sqrt{9 + 36}
\][/tex]
[tex]\[
= \sqrt{45}
\][/tex]
[tex]\[
\approx 6.71
\][/tex]
Hence, we have calculated that:
[tex]\[ KI \approx 4.12 \][/tex]
[tex]\[ KE \approx 4.12 \][/tex]
[tex]\[ IT \approx 6.71 \][/tex]
[tex]\[ TE \approx 6.71 \][/tex]
Since [tex]\( KI = KE \approx 4.12 \)[/tex] and [tex]\( IT = TE \approx 6.71 \)[/tex], KITE satisfies the condition of having two pairs of adjacent sides with equal lengths, confirming that it is indeed a kite.
Let's complete the proof in the form of the drop-down menu context given:
### Blank 1:
[tex]\[
\sqrt{(1-0)^2 + (2 - (-2))^2}
\][/tex]
[tex]\[
= \sqrt{(1)^2 + (4)^2}
\][/tex]
[tex]\[
= \sqrt{1 + 16}
\][/tex]
[tex]\[
= \sqrt{17}
\][/tex]
### Blank 2:
[tex]\[
\sqrt{17}
\][/tex]
### Blank 3:
[tex]\[
\sqrt{45}
\][/tex]
### Blank 4:
[tex]\[
\sqrt{45}
\][/tex]
Therefore, KITE is a kite because it has two pairs of adjacent sides that are equal: [tex]\( KI = KE \)[/tex] and [tex]\( IT = TE \)[/tex].
