To find the value of $x$ in Polygon Z, we can use the concept of similar polygons and scale factors. Since Polygon Z is a scaled copy of Polygon Y, there is a constant ratio between corresponding side lengths of the two polygons.

1. Identify the corresponding sides:

   • The side labeled $9/8$ in Polygon Y corresponds to the side labeled $3/4$ in Polygon Z.
   • The side labeled $9/10$ in Polygon Y corresponds to the side labeled $x$ in Polygon Z.

2. Determine the scale factor:
   To find the scale factor from Polygon Y to Polygon Z, divide the corresponding side lengths: $\text{Scale factor} = \frac{3/4}{9/8} = \frac{3 \cdot 8}{4 \cdot 9} = \frac{24}{36} = \frac{2}{3}$ So, the scale factor from Polygon Y to Polygon Z is $\frac{2}{3}$.

3. Apply the scale factor to find $x$:
   Since the scale factor is $\frac{2}{3}$, multiply the corresponding side in Polygon Y by $\frac{2}{3}$ to find $x$: $x = \frac{2}{3} \times \frac{9}{10} = \frac{18}{30} = \frac{3}{5}$

Therefore, the value of $x$ is $\frac{3}{5}$.

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:

1. How would the value of $x$ change if the scale factor was instead $\frac{3}{4}$?
2. If we knew only the lengths in Polygon Z, how could we find the lengths in Polygon Y?
3. How is the concept of similar polygons useful in real-world applications?
4. What happens to the area ratio of Polygon Y and Z if they are similar by a factor of $\frac{2}{3}$?
5. Can we use similar triangles within the polygons to verify the scale factor?

Tip: When working with similar polygons, remember that the ratio of side lengths remains consistent, but the areas scale by the square of the side length ratio.