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 = √((2 - (-2))²+(1 - 0)²)=√17, ke = √, it =, and te =. therefore, kite is a kite because

Explanation:

Step1: Recall 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 $IT$

For points $I(1,2)$ and $T(7,5)$, we have $x_1 = 1,y_1 = 2,x_2=7,y_2 = 5$. Then $IT=\sqrt{(7 - 1)^2+(5 - 2)^2}=\sqrt{6^{2}+3^{2}}=\sqrt{36 + 9}=\sqrt{45}=3\sqrt{5}$.

Step3: 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 $TE=\sqrt{(4 - 7)^2+(-1 - 5)^2}=\sqrt{(-3)^{2}+(-6)^{2}}=\sqrt{9 + 36}=\sqrt{45}=3\sqrt{5}$.

Step4: Recall kite - property

A kite has two pairs of adjacent sides that are equal in length. We already found $KI=\sqrt{(1 - 0)^2+(2+2)^2}=\sqrt{1 + 16}=\sqrt{17}$ and $KE=\sqrt{(4 - 0)^2+(-1 + 2)^2}=\sqrt{16 + 1}=\sqrt{17}$. Since $KI = KE$ and $IT=TE$, KITE is a kite.

Answer:

The blanks should be filled as follows:
$IT=\sqrt{(7 - 1)^2+(5 - 2)^2}=\sqrt{36 + 9}=\sqrt{45}$; $TE=\sqrt{(4 - 7)^2+(-1 - 5)^2}=\sqrt{9+36}=\sqrt{45}$; and we use the property that in a kite, two - pairs of adjacent sides are equal ( $KI = KE$ and $IT = TE$) to conclude that KITE is a kite.