To determine the value of $x$ in Polygon $R$, let's use the fact that $R$ is a scaled copy of $Q$. This means that the lengths in Polygon $R$ are proportional to those in Polygon $Q$.

Step-by-Step Solution:

1. Identify Corresponding Side Lengths:

   • In Polygon $Q$, we have side lengths 16, 14, and 6.
   • In Polygon $R$, the corresponding side lengths are 7, $x$, and 3.

2. Determine the Scale Factor:

   • We can find the scale factor by comparing the corresponding sides that we know in both polygons.
   • Let's take the side with lengths 14 in Polygon $Q$ and 7 in Polygon $R$.
   • Scale factor = $\frac{7}{14} = \frac{1}{2}$.

3. Apply the Scale Factor:

   • Since Polygon $R$ is a scaled copy of $Q$ with a scale factor of $\frac{1}{2}$, all corresponding side lengths in $R$ should be half of those in $Q$.
   • For the side with length 16 in $Q$, the corresponding side length in $R$ should be: $x = 16 \times \frac{1}{2} = 8$

Answer:

The value of $x$ is $8$.

Would you like further explanation or have any questions?

Here are five additional questions related to this problem:

1. How would the value of $x$ change if the scale factor were different?
2. What would be the value of $x$ if the length of the corresponding side in $Q$ was different?
3. How can we verify that $R$ is indeed a scaled copy of $Q$ using other side ratios?
4. What other methods could be used to find the value of $x$ without directly calculating the scale factor?
5. How does knowing the scale factor help in real-world applications of geometry?

Tip: When working with scaled figures, always identify at least one corresponding side pair to determine the scale factor accurately.