Question

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1. determine the length of \\( \overline { p q } \\).

\\( p q = \sqrt { ( - 4 - 2 ) ^ { 2 } + ( - 2 - 0 ) ^ { 2 } } \\)
\\( = \sqrt { ( \quad ) ^ { 2 } + ( \quad ) ^ { 2 } } \\)
\\( = \sqrt { ( \quad + \quad ) } \\)
\\( = \sqrt { \quad } \\)

1. determine whether \\( \overline { j k } \\) and \\( \overline { l m } \\) have the same length. justify your answer.

Explanation:

Step1: Calculate the values inside the square roots

For \(PQ=\sqrt{(-4 - 2)^2+(-2-0)^2}\), first calculate \(-4-2=-6\) and \(-2 - 0=-2\). So \(PQ=\sqrt{(-6)^2+(-2)^2}\).

Step2: Calculate the squares

\((-6)^2 = 36\) and \((-2)^2=4\). Then \(PQ=\sqrt{36 + 4}\).

Step3: Sum and simplify

\(36+4 = 40\), so \(PQ=\sqrt{40}=2\sqrt{10}\).

For \(JK\): Let \(J(-4,4)\) and \(K(-2,1)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(JK=\sqrt{(-2+4)^2+(1 - 4)^2}=\sqrt{2^2+(-3)^2}=\sqrt{4 + 9}=\sqrt{13}\).

For \(LM\): Let \(L(-1,-2)\) and \(M(2,1)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(LM=\sqrt{(2 + 1)^2+(1+2)^2}=\sqrt{3^2+3^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}\).

Since \(\sqrt{13}
eq\sqrt{18}\), \(JK\) and \(LM\) do not have the same length.

Answer:

1. \(PQ = 2\sqrt{10}\)
2. \(JK\) and \(LM\) do not have the same length.