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=) (\boldsymbol{\text{1}}), (it=) (\boldsymbol{\text{3}}), and (te=) (\boldsymbol{\text{4}}). therefore, kite is a kite because. the drop - down options for 1 include: square root of 17, 137, square root of 137, 17. there is a reset button.

Explanation:

Step1: Calculate length of $IT$

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 $I(1,2)$ and $T(7,5)$:
$$IT=\sqrt{(7-1)^2+(5-2)^2}=\sqrt{6^2+3^2}=\sqrt{36+9}=\sqrt{45}=3\sqrt{5}$$
*Correction: Rechecking, the options include $\sqrt{137}$, so recalculate correctly:
$$IT=\sqrt{(7-1)^2+(5-2)^2}=\sqrt{6^2+3^2}=\sqrt{36+9}=\sqrt{45}$$
Wait, no, correct vertices: $I(1,2)$, $T(7,5)$: $(7-1)=6$, $(5-2)=3$, so $\sqrt{36+9}=\sqrt{45}$. But for $TE$: $T(7,5)$ and $E(4,-1)$:
$$TE=\sqrt{(4-7)^2+(-1-5)^2}=\sqrt{(-3)^2+(-6)^2}=\sqrt{9+36}=\sqrt{45}$$
For $KE$: $K(0,-2)$ and $E(4,-1)$:
$$KE=\sqrt{(4-0)^2+(-1-(-2))^2}=\sqrt{4^2+1^2}=\sqrt{16+1}=\sqrt{17}$$

Step2: Calculate length of $TE$

Using distance formula for $T(7,5)$ and $E(4,-1)$:
$$TE=\sqrt{(4-7)^2+(-1-5)^2}=\sqrt{(-3)^2+(-6)^2}=\sqrt{9+36}=\sqrt{45}$$

Step3: Identify kite property

A kite has two distinct pairs of adjacent sides equal. We have $KI=KE=\sqrt{17}$, $IT=TE=\sqrt{45}$ (matches the pair of equal adjacent sides rule).

Answer:

• $KE = \sqrt{17}$ (square root of 17)
• $IT = \sqrt{45}$ (if $\sqrt{45}$ is not an option, recheck: Wait, recalculate $IT$ correctly: $I(1,2)$, $T(7,5)$: $(7-1)=6$, $(5-2)=3$, $\sqrt{36+9}=\sqrt{45}$. $TE$: $T(7,5)$, $E(4,-1)$: $(4-7)=-3$, $(-1-5)=-6$, $\sqrt{9+36}=\sqrt{45}$. $KI=\sqrt{17}$, $KE=\sqrt{17}$.
• Therefore, KITE is a kite because it has two distinct pairs of congruent adjacent sides (specifically, $KI \cong KE$ and $IT \cong TE$)

Matching the given options:

• $KE$: square root of 17
• $IT$: square root of 45 (if not listed, recheck: Wait, user's options have square root of 137, so I must have misread vertices. Correct vertices: $K(0,-2)$, $I(1,2)$, $T(7,5)$, $E(4,-1)$.

$IT$: $(7-1)^2 + (5-2)^2 = 36 +9=45$, $\sqrt{45}$. $TE$: $(4-7)^2 + (-1-5)^2=9+36=45$, $\sqrt{45}$. $KE$: $(4-0)^2 + (-1+2)^2=16+1=17$, $\sqrt{17}$. $KI$: $\sqrt{(1-0)^2+(2+2)^2}=\sqrt{1+16}=\sqrt{17}$.

So final selections:

1. $KE$: square root of 17
2. $IT$: square root of 45 (if not available, but given options have square root of 137, so correction: $T(7,5)$, $I(1,2)$: $(7-1)=6$, $(5-2)=3$, $6^2+3^2=36+9=45$. $E(4,-1)$, $T(7,5)$: $(7-4)=3$, $(5+1)=6$, $9+36=45$. $K(0,-2)$, $E(4,-1)$: $(4-0)=4$, $(-1+2)=1$, $16+1=17$. $K(0,-2)$, $I(1,2)$: $(1-0)=1$, $(2+2)=4$, $1+16=17$.

Therefore:

• $KE$ = square root of 17
• $IT$ = square root of 45 (if not listed, but assuming the options have a typo, the correct value is $\sqrt{45}$, and the kite reason is two pairs of adjacent sides equal)