To solve the problem, we need to analyze the given information involving triangles \( ABC \) and \( CDA \).
Given:
- Triangle \( ABC \) is congruent to triangle \( CDA \) by the Hypotenuse-Leg (HL) theorem, which applies to right triangles.
- \( AC = 21 , m \)
- \( AD = 29 , m \)
Since the triangles are congruent, we can deduce the following:
- Since \( AC \) is equal to \( 21 , m \) in triangle \( ABC \) and corresponds to side \( AD \) in triangle \( CDA \), this implies that the hypotenuse \( AC \) of triangle \( ABC \) also corresponds to \( AD = 29 , m \).
- Therefore, we can set up the conjecture that \( BA \) corresponds to the leg \( CD \).
- By the congruence of the triangles, we can write \( BA = CD \).
Now, since triangle congruences preserve the relationships between corresponding sides, and since we know \( AD \) is longer than \( AC \), we should look to find \( BA \) through some additional geometrical relationships, specifically the Pythagorean theorem.
However, given what you provided, we can't directly calculate \( BA \) without more information about the specific angles or other side lengths involved in these triangles, such as which sides correspond and whether there are right angles.
Let's summarize and assume both triangles are right triangles at points \( A \) and \( C \), respectively. Since we have established the lengths of the hypotenuses and the relationships, we would typically need to know more about the other sides to derive \( BA \).
If both \( AC \) and \( AD \) are the hypotenuses, and there's a right angle at points \( A \) and \( C \), we can also conclude that:
\[ AB^2 + BC^2 = AC^2 \quad \text{and} \quad CD^2 + AD^2 = AC^2. \]
However, without side lengths or angles, we cannot derive \( BA \). If you have additional information about angles or the relationship between the sides of triangles \( ABC \) and \( CDA \), please provide that for further assistance.