If ∆ABC CDA by the HL Theorem and AC = 84 m and AD 85 m, how long is BA?

Answer 1

In the context of right triangles, the Hypotenuse-Leg (HL) Theorem states that if two right triangles have equal hypotenuses and one pair of legs that are equal, then the triangles are congruent.

In your problem, you mentioned \( AC = 84 \) m and \( AD = 85 \) m. However, the triangle sides and how they correspond aren't entirely clear without more context or a visual aid.

If we assume that triangles \( ABC \) and \( CDA \) refer to two right triangles where:

• \( AC \) is the hypotenuse,
• \( AD \) is the leg of triangle \( CDA \),

and you want to find the length \( BA \), we need more information about the triangle structure or any additional triangle side lengths.

However, if we use a broader interpretation:

In right triangle \( ABC \):

• if we assume \( AC \) is a leg of triangle \( ABC \) (not the hypotenuse) and suppose \( AB \) is also one of the legs, we need information about the angle at \( A \) or the configuration of points to solve for \( BA \).

Please provide additional information or specify the relationships and configurations of your triangle sides so I can assist further!

Answer 2

13 m
13 m

22 m
22 m

27 m
27 m

15 m

Answer 3

It sounds like you are providing dimensions for multiple triangles or configurations, but it’s still unclear how they relate to your initial question about triangles \( ABC \) and \( CDA \) and the length of \( BA \).

If we assume you're looking for the lengths of sides in a specific right triangle in relation to the HL theorem, we essentially need to utilize the given dimensions.

However, if we break it down step by step, let’s examine the possible connections:

1. If \( AC = 84 , m \) and you meant \( AD = 85 , m \), it’s unclear how to connect them directly in terms of triangles without additional side information.

2. Given numbers such as 13 m, 22 m, 27 m, and 15 m could represent the other lengths of new triangles or sides.

Assuming you want to find a specific missing leg:

If we assume:

• \( AB = 13 , m \)
• \( BC = 22 , m \)
• And need to relate \( AC \) to these with Pythagorean theorem \( AB^2 + BC^2 = AC^2 \).

Then we derive:

• \( AC = \sqrt{AB^2 + BC^2} = \sqrt{(13, m)^2 + (22, m)^2} \)

Calculating:

\[ AC = \sqrt{169 + 484} = \sqrt{653} \approx 25.5 , m \]

However, your specific goal (to find \( BA \) or others) requires a clearer picture or a diagram of how the lengths are positioned relative to each triangle.

Please clarify the relationships between triangles and the specific question regarding \( BA \) with descriptions of each triangle or side lengths pertaining directly to \( BA \) so I could assist you in finding the needed lengths.