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=) square root of 17, (it = 3) (square root of 5), and (te = 3) (square root of 5). therefore, kite is a kite because one consecutive pair of adjacent sides have equal lengths the diagonals bisect each other two consecutive pairs of adjacent sides have equal lengths all sides have equal lengths

Explanation:

Step1: Calculate $KE$ with distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. For $K(0,-2)$ and $E(4,-1)$:
$$KE=\sqrt{(4-0)^2+(-1-(-2))^2}=\sqrt{16+1}=\sqrt{17}$$

Step2: Calculate $IT$ with distance formula

For $I(1,2)$ and $T(7,5)$:
$$IT=\sqrt{(7-1)^2+(5-2)^2}=\sqrt{36+9}=\sqrt{45}=3\sqrt{5}$$

Step3: Calculate $TE$ with distance formula

For $T(7,5)$ and $E(4,-1)$:
$$TE=\sqrt{(4-7)^2+(-1-5)^2}=\sqrt{9+36}=\sqrt{45}=3\sqrt{5}$$

Step4: Verify kite definition

A kite is defined as a quadrilateral with two consecutive pairs of adjacent sides of equal length. Here, $KI=KE=\sqrt{17}$ and $IT=TE=3\sqrt{5}$, which are two consecutive pairs of equal adjacent sides.

Answer:

$KE = \boldsymbol{\sqrt{17}}$
$IT = \boldsymbol{3\sqrt{5}}$
$TE = \boldsymbol{3\sqrt{5}}$
Therefore, KITE is a kite because: two consecutive pairs of adjacent sides have equal lengths