Final answer:
The given problem involves similar triangles, but the information provided is insufficient to answer the question. We need a full diagram or more context to establish the relationship between the triangles and solve for the unknown length, k.
Step-by-step explanation:
The question seems to be about similar triangles and involves calculating an unknown length using proportions. However, the given information is insufficient to provide a clear solution. As described, the question references various figures and distances but does not include enough context or a clear diagram to accurately complete the similarity statement or solve for k. For example, it's not clear how triangle AHTR relates to triangle AHR or how the distances given connect to each other.
To solve problems involving similar triangles, we need clear definitions of how the triangles are related, either by angle-side-angle (ASA), side-angle-side (SAS), or side-side-side (SSS) similarity. Once we establish similarity, we can use proportions to solve for unknown lengths, provided that the necessary information is given. Unfortunately, with the provided excerpts, a step-by-step explanation cannot be constructed.
If this question is derived from an accompanying textbook figure or previous discussion in class, it may be helpful to review that material or provide the full diagram for clarification. Otherwise, additional information would be needed to accurately perform calculations and find the value of k.