The two polygons shown in the image are similar trapezoids. Since they are similar, their corresponding sides are proportional.

Let’s set up a proportion using the corresponding sides:

• The longer base of the large trapezoid is 330 feet, and the corresponding base of the smaller trapezoid is 110 feet.
• The height of the larger trapezoid is 198 feet, and the height of the smaller trapezoid is 132 feet.
• The other base of the large trapezoid is 231 feet, and we need to solve for the corresponding base $x$ of the smaller trapezoid.

We can use the proportion:

$\frac{231}{x} = \frac{330}{110}$

Now, solve for $x$:

$\frac{231}{x} = 3 \quad (\text{since} \frac{330}{110} = 3)$

Multiply both sides by $x$ to isolate the variable:

$231 = 3x$

Now divide by 3:

$x = \frac{231}{3} = 77$

So, $x = 77$ feet.

Would you like more details or have any questions?

Here are some related questions to explore:

1. What is the definition of similar polygons?
2. How do you set up proportions in similar polygons?
3. Can this concept be applied to other shapes besides trapezoids?
4. What happens when the polygons are not similar? How do proportions change?
5. How can you check if two polygons are similar geometrically?

Tip: In problems involving similar figures, always identify corresponding sides carefully before setting up a proportion.