To solve this problem, let's examine the given information and apply properties of similar quadrilaterals.

1. Given:

   • Quadrilateral $JKLM$ is similar to quadrilateral $NOPQ$.
   • Side $JK = 8$ in $JKLM$ corresponds to side $NO = 39$ in $NOPQ$.
   • Side $JM = 13$ in $JKLM$ corresponds to side $QN$ in $NOPQ$, which we need to find.

2. Similarity Ratio: Since the quadrilaterals are similar, the ratios of corresponding sides are equal. Therefore, we can set up a proportion based on the sides we know: $\frac{JK}{NO} = \frac{JM}{QN}$ Substituting the known values: $\frac{8}{39} = \frac{13}{QN}$

3. Solving for $QN$: Cross-multiplying to solve for $QN$: $8 \cdot QN = 13 \cdot 39$ $8 \cdot QN = 507$ $QN = \frac{507}{8} = 63.375$

4. Rounding: Rounding $63.375$ to the nearest tenth, we get: $QN \approx 63.4$

Answer: The measure of side $QN$ is approximately 63.4.

Would you like more details or have any questions?

Here are some related questions to deepen understanding:

1. How does the similarity of polygons help in solving problems with unknown side lengths?
2. What are the other properties that are preserved in similar polygons besides side ratios?
3. How can we determine if two polygons are similar just by looking at side ratios?
4. What would change in the solution if a different pair of corresponding sides were used?
5. How can this concept of similarity be applied to triangles specifically?

Tip: When dealing with similar shapes, always confirm that corresponding sides are matched correctly to ensure accurate calculations.