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Determinar entre qué valores se
encuentra ac

To determine the possible range of values for the side \( AC \) in triangle \( ABC \), we can use the triangle inequality theorem. This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Given:
- \( AB = 5 \)
- \( BC = 4 \)
- Let \( AC = x \)

The triangle inequality theorem provides us with the following inequalities:

1. \( AB + BC > AC \)
2. \( AB + AC > BC \)
3. \( BC + AC > AB \)

Plugging in the known values:

1. \( 5 + 4 > x \)  
   \( 9 > x \)  
   \( x < 9 \)

2. \( 5 + x > 4 \)  
   \( x > 4 - 5 \)  
   \( x > -1 \)  
   (Since \( x \) represents the length of a side, it must be positive, so \( x > 0 \).)

3. \( 4 + x > 5 \)  
   \( x > 5 - 4 \)  
   \( x > 1 \)

From these inequalities, the only relevant range that satisfies all the conditions is:

\[
1 < x < 9
\]

Therefore, the length of side \( AC \) must be between \( 1 \) and \( 9 \).

Would you like more details or have any questions?

Here are five related questions:

1. What happens if \( x = 9 \) or \( x = 1 \)? Does it still form a valid triangle?
2. If another side of the triangle is known, can the angles be determined?
3. How does the triangle inequality theorem apply to right triangles?
4. If \( AC = 7 \), what would be the approximate angle opposite to side \( AC \)?
5. How does the Pythagorean theorem relate to the triangle inequality theorem?

**Tip:** Always remember that for any three segments to form a triangle, the sum of the lengths of any two sides must always be greater than the length of the third side.

Learn how to determine the possible range of values for side AC in triangle ABC using the Triangle Inequality Theorem. This problem involves applying the theorem to find constraints on the side lengths of a triangle. Suitable for students in Grades 7-9.

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