Question

here are the endpoints of the segments \\(\overline{jk}\\), \\(\overline{mn}\\), and \\(\overline{pq}\\).
\\( j(-2, 2), k(3, 4) \\)
\\( m(-6, 2), n(-7, 7) \\)
\\( p(-5, -8), q(-6, -2) \\)
follow the directions below.
(a) find the length of each segment.
give an exact answer (not a decimal approximation).
\\( jk = \square \\)
\\( mn = \square \\)
\\( pq = \square \\)
(b) check all statements that are true below.

Explanation:

Part (a)

To find the length of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\), we use the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

For \(JK\) with \(J(-2, 2)\) and \(K(3, 4)\):

Step 1: Identify coordinates

\(x_1 = -2\), \(y_1 = 2\), \(x_2 = 3\), \(y_2 = 4\)

Step 2: Apply distance formula

\[

$$\begin{align*} JK&=\sqrt{(3 - (-2))^2 + (4 - 2)^2}\\ &=\sqrt{(5)^2 + (2)^2}\\ &=\sqrt{25 + 4}\\ &=\sqrt{29} \end{align*}$$

\]

For \(MN\) with \(M(-6, 2)\) and \(N(-7, 7)\):

Step 1: Identify coordinates

\(x_1 = -6\), \(y_1 = 2\), \(x_2 = -7\), \(y_2 = 7\)

Step 2: Apply distance formula

\[

$$\begin{align*} MN&=\sqrt{(-7 - (-6))^2 + (7 - 2)^2}\\ &=\sqrt{(-1)^2 + (5)^2}\\ &=\sqrt{1 + 25}\\ &=\sqrt{26} \end{align*}$$

\]

For \(PQ\) with \(P(-5, -8)\) and \(Q(-6, -2)\):

Step 1: Identify coordinates

\(x_1 = -5\), \(y_1 = -8\), \(x_2 = -6\), \(y_2 = -2\)

Step 2: Apply distance formula

\[

$$\begin{align*} PQ&=\sqrt{(-6 - (-5))^2 + (-2 - (-8))^2}\\ &=\sqrt{(-1)^2 + (6)^2}\\ &=\sqrt{1 + 36}\\ &=\sqrt{37} \end{align*}$$

\]

Part (b)

Since the statements to check are not provided, we can't complete this part. But for part (a):

Answer:

(Part a):
\(JK = \boldsymbol{\sqrt{29}}\)
\(MN = \boldsymbol{\sqrt{26}}\)
\(PQ = \boldsymbol{\sqrt{37}}\)