Question

select the correct answers from each drop - down menu. complete the steps in the proof that show quadrilateral kite with vertices k(0, - 2), i(1,2), t(7,5), and e(4, - 1) is a kite. using the distance formula, ki = \\(\sqrt{(2-(-2))^{2}+(1 - 0)^{2}}=\sqrt{17}\\), ke = 17, it = 45, and te = 45. therefore, kite is a kite because one consecutive pair of adjacent sides have equal lengths all sides have equal lengths two consecutive pairs of adjacent sides have equal lengths the diagonals bisect each other

Explanation:

Step1: Recall the distance - formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Calculate $KI$

For points $K(0, - 2)$ and $I(1,2)$, we have $x_1 = 0,y_1=-2,x_2 = 1,y_2 = 2$. Then $KI=\sqrt{(1 - 0)^2+(2+2)^2}=\sqrt{1 + 16}=\sqrt{17}$.

Step3: Calculate $KE$

For points $K(0,-2)$ and $E(4,-1)$, we have $x_1 = 0,y_1=-2,x_2 = 4,y_2=-1$. Then $KE=\sqrt{(4 - 0)^2+(-1 + 2)^2}=\sqrt{16 + 1}=\sqrt{17}$.

Step4: Calculate $IT$

For points $I(1,2)$ and $T(7,5)$, we have $x_1 = 1,y_1 = 2,x_2 = 7,y_2 = 5$. Then $IT=\sqrt{(7 - 1)^2+(5 - 2)^2}=\sqrt{36+9}=\sqrt{45}$.

Step5: Calculate $TE$

For points $T(7,5)$ and $E(4,-1)$, we have $x_1 = 7,y_1 = 5,x_2 = 4,y_2=-1$. Then $TE=\sqrt{(4 - 7)^2+(-1 - 5)^2}=\sqrt{9 + 36}=\sqrt{45}$.

Step6: Recall the property of a kite

A kite is a quadrilateral with two pairs of adjacent sides that are equal in length. Here, $KI = KE=\sqrt{17}$ and $IT=TE=\sqrt{45}$.

Answer:

KITE is a kite because one consecutive pair of adjacent sides have equal lengths (i.e., $KI = KE$ and $IT = TE$).