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Complete the steps in the proof that show quadrilateral KPIE with vertices [tex]$K(0,-2), I(1,2), P(7,5)$[/tex], and [tex]$E(4,-1)$[/tex] is a kite.

Using the distance formula:
[tex]\[ K I = \sqrt{(2 - (-2))^2 + (1 - 0)^2} = \sqrt{17}, \][/tex]
[tex]\[ K E = \square, \][/tex]
[tex]\[ I P = \square, \][/tex]
[tex]\[ P E = \square. \][/tex]

Therefore, KPIE is a kite because [tex]$\square$[/tex].

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GinnyAnswer
To prove that the quadrilateral KITE is a kite using the distance formula, we'll go through each pair of adjacent sides and see if we can establish equal lengths for pairs of adjacent sides.

The coordinates of the vertices are given as:
[tex]\[ K (0, -2), \ I (1, 2), \ T (7, 5), \ E (4, -1) \][/tex]

First, let's use the distance formula to calculate the lengths of each side.

1. Length of KI:
[tex]\[ KI = \sqrt{(1-0)^2 + (2 - (-2))^2} = \sqrt{1 + 16} = \sqrt{17} \approx 4.123 \][/tex]

2. Length of KE:
[tex]\[ KE = \sqrt{(4-0)^2 + (-1 - (-2))^2} = \sqrt{16 + 1} = \sqrt{17} \approx 4.123 \][/tex]

3. Length of IT:
[tex]\[ IT = \sqrt{(7-1)^2 + (5-2)^2} = \sqrt{36 + 9} = \sqrt{45} \approx 6.708 \][/tex]

4. Length of TE:
[tex]\[ TE = \sqrt{(4-7)^2 + (-1-5)^2} = \sqrt{9 + 36} = \sqrt{45} \approx 6.708 \][/tex]

Therefore, we have the lengths:
- [tex]\( KI \approx 4.123 \)[/tex]
- [tex]\( KE \approx 4.123 \)[/tex]
- [tex]\( IT \approx 6.708 \)[/tex]
- [tex]\( TE \approx 6.708 \)[/tex]

So, we can see that [tex]\( KI = KE \)[/tex] and [tex]\( IT = TE \)[/tex]. This means the quadrilateral has two distinct pairs of adjacent sides that are equal.

Therefore, KITE is a kite because it has two pairs of adjacent sides that are equal.

Filling in the blanks of the given proof:

Using the distance formula,
[tex]\[ KI = \sqrt{(2-(-2))^2 + (1-0)^2} = \sqrt{17}, \][/tex]
[tex]\[ KE = \sqrt{17}, \][/tex]
[tex]\[ IT = \sqrt{45}, \][/tex]
[tex]\[ TE = \sqrt{45}. \][/tex]

Therefore, KITE is a kite because it has two pairs of adjacent sides that are equal.
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