To solve this problem, we can use the fact that the two triangles are similar. For similar triangles, corresponding sides are proportional.

Step-by-Step Process:

1. The corresponding side lengths of the two triangles are proportional because the triangles are similar.

   Let's define the triangles as follows:

   • In the smaller triangle:
     • One known side is 15 inches, and the other is $j$.
   • In the larger triangle:
     • The corresponding sides are 21 inches and 14 inches, respectively.

2. Set up a proportion: $\frac{\text{side of small triangle}}{\text{corresponding side of large triangle}} = \frac{\text{other side of small triangle}}{\text{corresponding other side of large triangle}}$ Using the known values: $\frac{j}{14} = \frac{15}{21}$

3. Solve for $j$: $j = \frac{15}{21} \times 14 = \frac{5}{7} \times 14 = 10 \, \text{inches}$

Thus, the missing length $j$ is 10 inches.

Would you like further details or clarification on any step?

Related Questions:

1. How do you determine if two triangles are similar?
2. Can this method be applied to non-triangular shapes?
3. What happens if the triangles are not similar?
4. How do you find the scale factor between two similar triangles?
5. What are the criteria for triangle similarity?

Tip: Always ensure the sides you are comparing are corresponding ones by checking their relative positions (angles can help identify this).