Question
If triangle ABC is congruent to triangle CDA by the HL theorem and AC = 21cm and AD = 29cm, how long is BA?
Answers
Answer
its 20cm
Answered by
GPT-4o mini
Since triangle ABC is congruent to triangle CDA by the Hypotenuse-Leg (HL) theorem, we have the following correspondences based on the congruence of the triangles:
- \( AC \) corresponds to \( AD \)
- \( AB \) corresponds to \( CD \)
- \( BC \) corresponds to \( CA \)
Given the lengths:
- \( AC = 21 \, \text{cm} \)
- \( AD = 29 \, \text{cm} \)
Since the triangles are congruent, the corresponding sides are equal:
- \( AB = CD \)
- \( AC = AD \)
However, it seems that I have made a mistake in interpreting the correspondence. Since they are congruent on HL basis, it means that:
- \( AC \) is the hypotenuse in triangle ABC and should correspond to the hypotenuse in triangle CDA. Hence, it should match with the length of the triangle where CD is the leg, not hypotenuse.
So we don't directly see CD here, but we realize that legs will be equal on both sides, which makes us understand that:
To find \( BA \), we consider the corresponding lengths. Since none are given, and using the property of congruence leads us to infer that segments \( BA \) (or \( AB \)) will have relations but can't be solved with the current data.
Thus with AC and AD lengths available, we cannot find \( BA \) because it requires additional information regarding other sides or angles. The provided lengths \( AC \) and \( AD \) alone do not determine \( BA \). If more details are included, a completion can be made, else it can remain unknown.
Therefore, **BA cannot be determined with the given information.**
- \( AC \) corresponds to \( AD \)
- \( AB \) corresponds to \( CD \)
- \( BC \) corresponds to \( CA \)
Given the lengths:
- \( AC = 21 \, \text{cm} \)
- \( AD = 29 \, \text{cm} \)
Since the triangles are congruent, the corresponding sides are equal:
- \( AB = CD \)
- \( AC = AD \)
However, it seems that I have made a mistake in interpreting the correspondence. Since they are congruent on HL basis, it means that:
- \( AC \) is the hypotenuse in triangle ABC and should correspond to the hypotenuse in triangle CDA. Hence, it should match with the length of the triangle where CD is the leg, not hypotenuse.
So we don't directly see CD here, but we realize that legs will be equal on both sides, which makes us understand that:
To find \( BA \), we consider the corresponding lengths. Since none are given, and using the property of congruence leads us to infer that segments \( BA \) (or \( AB \)) will have relations but can't be solved with the current data.
Thus with AC and AD lengths available, we cannot find \( BA \) because it requires additional information regarding other sides or angles. The provided lengths \( AC \) and \( AD \) alone do not determine \( BA \). If more details are included, a completion can be made, else it can remain unknown.
Therefore, **BA cannot be determined with the given information.**