To determine if 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, we will use the distance formula to find the lengths of its sides.
The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
1. Finding [tex]\(KI\)[/tex]:
[tex]\[
KI = \sqrt{(1 - 0)^2 + (2 + 2)^2} = \sqrt{1^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17} \approx 4.123
\][/tex]
2. Finding [tex]\(KE\)[/tex]:
[tex]\[
KE = \sqrt{(4 - 0)^2 + (-1 + 2)^2} = \sqrt{4^2 + (-1 + 2)^2} = \sqrt{16 + 1} = \sqrt{17} \approx 4.123
\][/tex]
3. Finding [tex]\(IT\)[/tex]:
[tex]\[
IT = \sqrt{(7 - 1)^2 + (5 - 2)^2} = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} \approx 6.708
\][/tex]
4. Finding [tex]\(TE\)[/tex]:
[tex]\[
TE = \sqrt{(7 - 4)^2 + (5 + 1)^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} \approx 6.708
\][/tex]
Thus, we obtain:
[tex]\[
KI \approx 4.123, \quad KE \approx 4.123, \quad IT \approx 6.708, \quad TE \approx 6.708
\][/tex]
5. Determining if [tex]\(KITE\)[/tex] is a kite:
A quadrilateral is a kite if it has two distinct pairs of adjacent sides that are equal. From the calculated distances, we see:
[tex]\[
KI = KE \quad \text{and} \quad IT = TE
\][/tex]
Therefore, quadrilateral [tex]\(KITE\)[/tex] is a kite because it has two pairs of adjacent sides that are equal.
Here are the completed steps:
- [tex]\(KE = \sqrt{17}\)[/tex]
- [tex]\(IT = \sqrt{45}\)[/tex]
- [tex]\(TE = \sqrt{45}\)[/tex]
- Therefore, [tex]\(KITE\)[/tex] is a kite because [tex]\(KI = KE\)[/tex] and [tex]\(IT = TE\)[/tex]
