Explanation:

- In triangle ABC, DE is parallel to BC.

- When DE is parallel to BC, the triangles ADE and ABC are similar by the Basic Proportionality Theorem (also known as Thales's Theorem).

- The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides, then it divides those sides in the same ratio.

- Given: AD : DB = 9 : 8.

- Because DE is parallel to BC, DE : BC is the same as AD : AB.

- AD : AB is equivalent to AD : (AD + DB).

- Therefore, AD : AB = 9 : (9 + 8) = 9 : 17.

- Hence, DE : BC = 9 : 17.

Option 1: 9 : 17 is the correct answer.