This problem involves similar polygons, as polygon $B$ is a scaled copy of polygon $A$. Since they are similar, the ratio of corresponding sides between the two polygons remains the same.

Given:

• Side lengths of Polygon $A$: 15, 35, and 20.
• Side lengths of Polygon $B$: 6, $c$, and 8.

We will find the scale factor between the two polygons by comparing corresponding side lengths.

Step 1: Find the scale factor.

The side corresponding to 15 in Polygon $A$ is 6 in Polygon $B$. Thus, the scale factor $k$ is:

$k = \frac{6}{15} = \frac{2}{5}$

Step 2: Find the value of $c$.

To find the value of $c$, we use the fact that the corresponding side in Polygon $A$ is 35. Since the scale factor is $\frac{2}{5}$, we can set up the proportion:

$\frac{c}{35} = \frac{2}{5}$

Step 3: Solve for $c$.

$c = 35 \times \frac{2}{5} = 14$

Thus, the value of $c$ is $14$.

Would you like more details on how we used similarity in this problem?

Here are 5 related questions for further exploration:

1. If the area of Polygon $A$ is given, how would you find the area of Polygon $B$?
2. What is the perimeter of Polygon $B$ if the perimeter of Polygon $A$ is known?
3. How do the angles of two similar polygons compare?
4. Can you calculate the scale factor by comparing the height (20 and 8) instead of the base?
5. If polygon $B$ had a different scale factor, how would it affect the side lengths?

Tip: In similar polygons, not only the sides but also the ratios of areas follow a squared relationship with the scale factor!