Short Answer 
The problem involves two similar triangles, ‚ñ≥ABC and ‚ñ≥BDC, where the equality of certain angles confirms their similarity. After establishing a proportional relationship and solving for the length of side BC, which is 12, the similarity scale is found to be 3/4, leading to the relationship between the sides of the triangles.
Step 1: Identify Similar Triangles
In this problem, we have two triangles, △ABC and △BDC, where point D lies on line AC. The angles, specifically m∠BDC and m∠ABC, are equal, indicating that these triangles are similar by the AA (Angle-Angle) rule. This similarity leads to the application of the proportionality theorem, which allows us to set up a ratio based on the corresponding sides of these triangles.
Step 2: Set Up Proportional Relationships
We define the lengths involved where AD is 7 and DC is 9. To find the length of BC, we denote it as x. The proportionality relationship from the similarity of triangles gives us the equation:
- BC / AC = DC / BC
- x / 16 = 9 / x
By cross-multiplying and simplifying, we can solve the equation to find that x equals 12, hence BC = 12.
Step 3: Determine the Similarity Scale
After finding the length of BC, we proceed to determine the similarity scale k. This is calculated as:
- k = BC / AC = 12 / 16 = 3 / 4
Using this scale, we relate the sides of triangles. From the proportion, we deduce that:
- BD / BA = 3 / 4
- 3BA = 4BD
- Thus, BA = (3/4)BD