In triangles ΔABC and ΔCDA, which are congruent by the Hypotenuse-Leg (HL) Theorem, we can conclude the following:
-
Given that AC = 84 m and AD = 85 m, we identify that in right triangles, the hypotenuses AC and AD correspond to the right angle in ΔABC and ΔCDA, respectively.
-
The sides adjacent to these hypotenuses are the legs of the triangles. We can use the Pythagorean theorem to determine the lengths of the other sides.
Let's denote:
- \( BA \) as \( x \)
- \( BC \) and \( CD \) will be equal because of the congruence of the triangles.
Using the Pythagorean theorem for triangle ΔABC:
\[ AC^2 = AB^2 + BC^2 \]
And for triangle ΔCDA:
\[ AD^2 = CD^2 + AC^2 \]
Since \( CD = BC \) (due to congruence), we can set \( x = BA \) and \( y = BC = CD \).
For ΔABC:
\[ 84^2 = x^2 + y^2 \quad (1) \]
For ΔCDA:
\[ 85^2 = y^2 + AC^2 \quad (2) \]
Plugging in the values:
From equation (1):
\[ 84^2 = x^2 + y^2 \implies 7056 = x^2 + y^2 \]
From equation (2):
\[ 85^2 = y^2 + 84^2 \implies 7225 = y^2 + 7056 \]
So,
\[ y^2 = 7225 - 7056 = 169 \implies y = 13 \]
Now, inserting \( y = 13 \) back into equation (1):
\[ 7056 = x^2 + 13^2 \] \[ 7056 = x^2 + 169 \] \[ x^2 = 7056 - 169 = 6887 \] \[ x = \sqrt{6887} \approx 83 \]
Therefore, the only side we need is BA, which does not help derive the correct answer, as it seems I made this more complex than required.
Returning back to the given:
Given AC and AD with lengths 84m and 85m, the possible answers indicate a simpler relation based on the congruency.
Since both triangles are congruent (structure is similar), and we established lengths corresponding to congruence, the congruent side opposite to the hypotenuse in triangle might lead to the indication of BA = 13m in the simplest congruence tied to the HL theorem.
Thus, the length of \( BA \) is:
13 m.