To determine the length of \( \overline{BA} \) in the given congruent triangles \( \Delta ABC \) and \( \Delta CDA \), we can utilize the properties of congruent triangles.
Since the triangles are congruent (\( \Delta ABC \cong \Delta CDA \)), corresponding sides are equal. In this case, we can identify the corresponding sides from the congruence:
- \( AC \) corresponds to \( CA \) (which is the same segment, just different naming)
- \( AB \) corresponds to \( CD \)
- \( BC \) corresponds to \( DA \)
You provided the lengths of \( AC \) and \( AD \):
- \( AC = 84 , \text{m} \)
- \( AD = 85 , \text{m} \)
Since \( CA \) and \( AC \) are the same, we move on to \( AB \) and \( CD \). We need to find \( AB \) (which is \( \overline{BA} \)).
Since the triangles are congruent:
- If \( AB = CD \), then \( CD \) must also be \( 84 , \text{m} \) (since \( AC \) = \( 84 , \text{m} \)).
- From \( AD = 85 , \text{m} \), but since \( AD \) corresponds directly to \( DA \), we also understand \( AD \) doesn’t play a direct role in determining \( AB \).
Therefore, \( AB = 85 , \text{m} \) because it corresponds with the congruent sides established through the HL (Hypotenuse-Leg) theorem.
Thus, the length of \( \overline{BA} \) is \( 85 , \text{m} \).