We are given two quadrilaterals representing pens, and the question is asking how much fence Paul will need to finish the larger pen if the two shapes are similar.

Steps to solve:

1. Perimeter Ratio: Since the shapes are similar, the ratio of corresponding side lengths between the two pens is constant.

2. Find the scale factor:

   • Look at two corresponding sides of the smaller and larger pens. For example:
     • In the smaller pen, $AB = 40 \, \text{ft}$ and the corresponding side $PQ = 120 \, \text{ft}$ in the larger pen.
     • The scale factor is $\frac{PQ}{AB} = \frac{120}{40} = 3$.

3. Calculate the perimeter of the smaller pen:

   • The sides of the smaller pen are $AB = 40 \, \text{ft}, BC = 80 \, \text{ft}, CD = 100 \, \text{ft}, AD = 60 \, \text{ft}$.
   • The total perimeter of the smaller pen is: $P_{\text{small}} = 40 + 80 + 100 + 60 = 280 \, \text{ft}$

4. Find the perimeter of the larger pen:

   • Since the pens are similar, the perimeter of the larger pen is scaled by the same factor of 3: $P_{\text{large}} = 280 \times 3 = 840 \, \text{ft}$

5. Determine how much more fence Paul needs:

   • Paul already has the perimeter of the smaller pen, so the additional fence needed for the larger pen is: $\text{Additional fence} = P_{\text{large}} - P_{\text{small}} = 840 - 280 = 560 \, \text{ft}$

Thus, the correct answer should be D. 280 ft.

Would you like more details or have further questions?

Here are 5 related questions you might find useful:

1. How do you determine whether two shapes are similar?
2. What is the relationship between scale factors and areas of similar shapes?
3. How would you calculate the area of each pen if they are similar shapes?
4. Can this concept of scale factors be applied to three-dimensional figures?
5. What happens to the ratio of perimeters if the scale factor changes?

Tip: When working with similar figures, always remember that the ratio of their areas is the square of the ratio of their side lengths!