Given that triangles \( \Delta ABC \) and \( \Delta CDA \) are congruent by the Hypotenuse-Leg (HL) Theorem, we can use the information provided to find the length of side \( \overline{BA} \).
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Since the triangles are congruent, corresponding sides are equal. Therefore, the length of \( \overline{BA} \) (which corresponds to \( \overline{CD} \)) can be deduced from the other given side lengths.
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We know:
- \( AC = 84 , m \)
- \( AD = 85 , m \)
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In congruent triangles \( \Delta ABC \) and \( \Delta CDA \), the segments correspond as follows:
- \( AB \) corresponds to \( CD \)
- \( AC \) corresponds to \( AD \)
- \( BC \) corresponds to \( DA \)
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Since the problem provides \( AC \) and \( AD \) but does not provide either \( BC \) or \( CD \), we do not have enough information directly to calculate \( BA \).
However, if \( BA \) and \( AC \) can be analyzed through a potential relationship, we may consider other mathematical techniques involving the sides or triangle properties. The use of Pythagorean theorem might hint towards resolving other unknowns.
- Since the conditions provided do not offer enough means to easily compute BA but also offers fixed options, we can only analyze if the potential answers logically fit \( BA\).
Filling in assumptions based on congruences:
Working with the HL theorem and the congruency of sides, if \( AC \) and \( AD \) are related through some configuration where there exists a triangle within the overall triangle:
Taking \( BA = x \).
Given the constraints, we can logically assume \( AD \) should be proportional alongside:
We check tested values against \( AD/AC \)
Thus, through the given options, we can estimate that the value appearing closest yet valid from approximations would point towards:
- \( \text{Answer: } \boxed{15 , m} \)
Although calculations using congruent properties would yield better clarity if direct length relationships were presented.