Question

14

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 =\\) choose an answer, and \\(te =\\) 45.
therefore, choose an answer because two consecutive pairs of adjacent sides have equal lengths.

Explanation:

⚡ Using what you learned: transformations in the coordinate plane

Step 1: Calculate KE

Using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) for \( K(0,-2) \) and \( E(4,-1) \):

\[ KE = \sqrt{(4 - 0)^2 + (-1 - (-2))^2} \]
\[ KE = \sqrt{4^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17} \]

So, \( KE = \text{square root of 17} \).

Step 2: Calculate IT

Using the distance formula for \( I(1,2) \) and \( T(7,5) \):

\[ IT = \sqrt{(7 - 1)^2 + (5 - 2)^2} \]
\[ IT = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} \]

Simplifying the radical:
\[ \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} \]

So, \( IT = \text{3 (square root of 5)} \).

Step 3: Calculate TE

Using the distance formula for \( T(7,5) \) and \( E(4,-1) \):

\[ TE = \sqrt{(4 - 7)^2 + (-1 - 5)^2} \]
\[ TE = \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} \]

Simplifying the radical:
\[ \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} \]

So, \( TE = \text{3 (square root of 5)} \) (or \( \sqrt{45} \) depending on the dropdown option).

Step 4: Complete the proof statement

Since \( KI = KE = \sqrt{17} \) and \( IT = TE = 3\sqrt{5} \), quadrilateral \( KITE \) is a kite because two consecutive pairs of adjacent sides have equal lengths.

Answer:

• \( KE \): square root of 17
• \( IT \): 3 (square root of 5)
• \( TE \): 3 (square root of 5) (or 45 under the square root, which simplifies to 3 (square root of 5))
• Therefore, \( KITE \) is a kite because: two consecutive pairs of adjacent sides have equal lengths