Explanation:

Let’s break down the problem:

- D and E are on AB and AC, and DE ? BC, which means triangles ADE and ABC are similar by the Basic Proportionality Theorem.

- You’re told AD : DB = 7 : 9. So, AD/DB = 7/9.

- That means DE divides the sides in the same ratio, so AE/EC = 7/9 too.

- The ratio of the areas of similar triangles is the square of the similarity ratio. So, area(?ADE)/area(?ABC) = (7/16)² = 49/256.

- But the question asks about ?DEF and ?CBF. F is the intersection of CD and BE.

Let’s talk about the options:

1. 49 : 144 – This would be correct only if the sides' ratios were 7/12.

2. 49 : 81 – This would fit if the sides were divided in 7:9 and comparing corresponding triangles directly.

3. 49 : 256 – That’s the square of 7/16.

4. 256 : 49 – This just flips the previous ratio.

But here’s the thing: We’re asked about area(?DEF)/area(?CBF), not area(?ADE)/area(?ABC). For ?DEF and ?CBF, both are kinds of "cut out" triangles, but from the proportionality, it turns out that the ratio ?DEF : ?CBF is (AD/AB)² = (7/16)² = 49/256.

So the correct option is:

- Option 3: 49 : 256

In summary:

- The two triangles are similar by the basic proportionality theorem.

- Area ratios go as square of the side ratios, specifically (7/16)².

- Option 3 matches that exactly: 49 : 256.