Let's go through the steps to show that quadrilateral KITE is a kite using the distance formula.
We need to calculate the lengths of KI, KE, IT, and TE using the coordinates given for K (0,-2), I (1,2), T (7,5), and E (4,-1).
1. Calculate [tex]\( KI \)[/tex]:
Using the coordinates K (0,-2) and I (1,2):
[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]:
Using the coordinates K (0,-2) and E (4,-1):
[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]:
Using the coordinates I (1,2) and T (7,5):
[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]:
Using the coordinates T (7,5) and E (4,-1):
[tex]\[
TE = \sqrt{(7 - 4)^2 + (5 - (-1))^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} \approx 6.708
\][/tex]
Now, let's fill in the blanks:
Using the distance formula, [tex]\( KI = \sqrt{17} \)[/tex], [tex]\( KE = \sqrt{17} \)[/tex], [tex]\( IT = \sqrt{45} \)[/tex], and [tex]\( TE = \sqrt{45} \)[/tex].
Therefore, KITE is a kite because it has two pairs of adjacent sides that are equal in length: [tex]\( KI = KE \)[/tex] and [tex]\( IT = TE \)[/tex].
