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

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**(a) Finding the lengths using the distance formula**

The distance formula between two points $$ (x_1, y_1) $$ and $$ (x_2, y_2) $$ is given by:
$$
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

1. For $$\overline{JK}$$ with $$ J(3,1) $$ and $$ K(-2,-5) $$:
   $$
   \begin{aligned}
   JK &= \sqrt{(-2 - 3)^2 + (-5 - 1)^2} \
      &= \sqrt{(-5)^2 + (-6)^2} \
      &= \sqrt{25 + 36} \
      &= \sqrt{61}
   \end{aligned}
   $$

2. For $$\overline{MN}$$ with $$ M(-8,4) $$ and $$ N(-4,-3) $$:
   $$
   \begin{aligned}
   MN &= \sqrt{(-4 - (-8))^2 + (-3 - 4)^2} \
      &= \sqrt{(4)^2 + (-7)^2} \
      &= \sqrt{16 + 49} \
      &= \sqrt{65}
   \end{aligned}
   $$

3. For $$\overline{PQ}$$ with $$ P(-7,8) $$ and $$ Q(-2,2) $$:
   $$
   \begin{aligned}
   PQ &= \sqrt{(-2 - (-7))^2 + (2 - 8)^2} \
      &= \sqrt{(5)^2 + (-6)^2} \
      &= \sqrt{25 + 36} \
      &= \sqrt{61}
   \end{aligned}
   $$

Thus, the exact lengths are:
$$
\begin{array}{l}
JK=\sqrt{61} \
MN=\sqrt{65} \
PQ=\sqrt{61}
\end{array}
$$

**(b) Checking which segments are congruent**

Two segments are congruent if they have the same length.

- $$\overline{JK}$$ has length $$\sqrt{61}$$.
- $$\overline{MN}$$ has length $$\sqrt{65}$$.
- $$\overline{PQ}$$ has length $$\sqrt{61}$$.

Comparing the lengths:
- $$\overline{JK} \cong \overline{PQ}$$ because both are $$\sqrt{61}$$.
- $$\overline{MN}$$ is not congruent to either $$\overline{JK}$$ or $$\overline{PQ}$$ because $$\sqrt{65} \neq \sqrt{61}$$.

The only true statement is:
$$
\overline{JK} \cong \overline{PQ}
$$
**(a) Lengths of the Segments** $$ \begin{array}{l} JK = \sqrt{61} \ MN = \sqrt{65} \ PQ = \sqrt{61} \end{array} $$ **(b) Congruent Segments** $$ \overline{JK} \cong \overline{PQ} $$

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