Here are the endpoints of the segments \( \overline{J K}, \overline{M N} \), and \( \overline{P Q} \). \[ \begin{array}{l} J(3,1), K(-2,-5) \\ M(-8,4), N(-4,-3) \\ P(-7,8), Q(-2,2) \end{array} \] Follow the directions below. (a) Find the length of each segment. Give an exact answer (not a decimal approximation). \[ \begin{array}{l} J K=\square \\ M N=\square \\ P Q=\square \end{array} \] (b) Check all statements that are true below. \( \overline{J K} \cong \overline{M N} \) \( \overline{J K} \cong \overline{P Q} \) \( \overline{M N} \cong \overline{P Q} \) None of these are true

To find the lengths of the segments \( \overline{J K} \), \( \overline{M N} \), and \( \overline{P Q} \), we can use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] (a) Let's calculate each segment's length step by step. 1. **For segment \( \overline{J K} \)** with points \( J(3,1) \) and \( K(-2,-5) \): \[ d_{JK} = \sqrt{((-2) - 3)^2 + ((-5) - 1)^2} = \sqrt{(-5)^2 + (-6)^2} = \sqrt{25 + 36} = \sqrt{61} \] Thus, \( J K = \sqrt{61} \). 2. **For segment \( \overline{M N} \)** with points \( M(-8,4) \) and \( N(-4,-3) \): \[ d_{MN} = \sqrt{((-4) - (-8))^2 + ((-3) - 4)^2} = \sqrt{(4)^2 + (-7)^2} = \sqrt{16 + 49} = \sqrt{65} \] Thus, \( M N = \sqrt{65} \). 3. **For segment \( \overline{P Q} \)** with points \( P(-7,8) \) and \( Q(-2,2) \): \[ d_{PQ} = \sqrt{((-2) - (-7))^2 + ((2) - 8)^2} = \sqrt{(5)^2 + (-6)^2} = \sqrt{25 + 36} = \sqrt{61} \] Thus, \( P Q = \sqrt{61} \). So, the lengths of the segments are: \[ \begin{array}{l} J K=\sqrt{61} \\ M N=\sqrt{65} \\ P Q=\sqrt{61} \end{array} \] (b) Now let's check the statements: 1. \( \overline{J K} \cong \overline{M N} \) – False, since \( \sqrt{61} \neq \sqrt{65} \). 2. \( \overline{J K} \cong \overline{P Q} \) – True, since both lengths are \( \sqrt{61} \). 3. \( \overline{M N} \cong \overline{P Q} \) – False, since \( \sqrt{65} \neq \sqrt{61} \). Thus, the true statement is: \( \overline{J K} \cong \overline{P Q} \).