Certainly! Let's complete the steps to prove that the quadrilateral [tex]\( KITE \)[/tex] 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.
Using the distance formula, the distances between the points are calculated as follows:
1. Calculate [tex]\( KI \)[/tex]:
[tex]\[
KI = \sqrt{(I_x - K_x)^2 + (I_y - K_y)^2}
= \sqrt{(1 - 0)^2 + (2 - (-2))^2}
= \sqrt{1 + 16}
= \sqrt{17}
\][/tex]
2. Calculate [tex]\( KE \)[/tex]:
[tex]\[
KE = \sqrt{(E_x - K_x)^2 + (E_y - K_y)^2}
= \sqrt{(4 - 0)^2 + (-1 - (-2))^2}
= \sqrt{16 + 1}
= \sqrt{17}
\][/tex]
3. Calculate [tex]\( IT \)[/tex]:
[tex]\[
IT = \sqrt{(T_x - I_x)^2 + (T_y - I_y)^2}
= \sqrt{(7 - 1)^2 + (5 - 2))^2
= \sqrt{36 + 9}
= \sqrt{45}
\][/tex]
4. Calculate [tex]\( ET \)[/tex]:
[tex]\[
ET = \sqrt{(T_x - E_x)^2 + (T_y - E_y)^2}
= \sqrt{(7 - 4)^2 + (5 + 1))^2}
= \sqrt{9 + 36}
= \sqrt{45}
\][/tex]
Summarizing the results, we have the lengths:
[tex]\[
KI = \sqrt{17}, \quad KE = \sqrt{17}, \quad IT = \sqrt{45}, \quad ET = \sqrt{45}
\][/tex]
Both pairs of adjacent sides [tex]\( KI \)[/tex] and [tex]\( KE \)[/tex] as well as [tex]\( IT \)[/tex] and [tex]\( ET \)[/tex] are equal in length.
Therefore, [tex]\( KITE \)[/tex] is a kite because it has two pairs of adjacent sides of equal length.
Now, let's fill in the drop-down menu selections:
- [tex]\(\sqrt{17}\)[/tex]
- [tex]\(KE = \sqrt{17}\)[/tex]
- [tex]\(IT = \sqrt{45}\)[/tex]
- [tex]\(ET = \sqrt{45}\)[/tex]
- [tex]\(Therefore, \ KITE \ is \ a \ kite \ because \ it \ has \ two \ pairs \ of \ adjacent \ sides \ of \ equal \ length.\)[/tex]
