Explanation:

Let’s break it down:

- You’ve got triangle ABC. D is on AB. E and F are on BC.

- DF is parallel to AC, and DE is parallel to AF. These clues tell you we’re dealing with similar triangles and proportional segments because of parallel lines.

- BE = 4 cm, EF = 6 cm, and you’re asked for the full length BC.

Here’s what matters:

- Since DE is parallel to AF, the segments on BC (BE and EF) are in geometric progression.

- Let’s put it together: if BE = 4 and EF = 6, the ratio of BE:EF = 4:6 = 2:3.

- If we say the next segment (let’s call it FC) is x, the sequence is 4 (BE), 6 (EF), x (FC). By the properties of similar triangles (and geometric progression due to parallel lines), BE : EF = EF : FC, so 4/6 = 6/x ? x = 9.

- Add it up: BC = BE + EF + FC = 4 + 6 + 9 = 19.

Now the options:

- 1: 30

- 2: 20

- 3: 25

- 4: 15

None match exactly with 19, but the closest, based on the ratio tricks, would have been option 2 (20), not 25.

That’s the situation. The trick is recognizing the sequence is geometric, not arithmetic. 19 cm (closest to 20 cm) is the right ballpark.