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Answer :
Sure, let's solve this step-by-step using the triangle inequality theorem.
### Triangle Inequality Theorem
The triangle inequality theorem states that for any triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
- [tex]\(a + b > c\)[/tex]
- [tex]\(a + c > b\)[/tex]
- [tex]\(b + c > a\)[/tex]
Given:
- [tex]\(a = 29\)[/tex]
- [tex]\(b = 40\)[/tex]
Let's determine the constraints for the third side, [tex]\(c\)[/tex].
### Step-by-Step Calculation:
1. Determine the upper bound for [tex]\(c\)[/tex]:
According to the inequality theorem, [tex]\(a + b > c\)[/tex]. So,
[tex]\[ 29 + 40 > c \][/tex]
[tex]\[ 69 > c \][/tex]
This means that [tex]\(c\)[/tex] must be less than 69.
2. Determine the lower bound for [tex]\(c\)[/tex]:
We need to consider both [tex]\( |a - b| < c\)[/tex] and [tex]\(a \neq b \)[/tex], but since the specific constraint is more restrictive, it is more effective to use:
[tex]\[ |29 - 40| < c \][/tex]
This absolute difference is important:
[tex]\[ 11 < c \][/tex]
This means that [tex]\(c\)[/tex] must be greater than 11.
### Combining the Results:
From the above two points, the third side [tex]\(c\)[/tex] must satisfy:
[tex]\[ 11 < c < 69 \][/tex]
### Conclusion:
The correct range in which the length of the third side must lie is:
[tex]\[ 11 < x < 69 \][/tex]
Thus, option B and C are both correct.
[tex]\[
\boxed{11 < x < 69}
\][/tex]
### Triangle Inequality Theorem
The triangle inequality theorem states that for any triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
- [tex]\(a + b > c\)[/tex]
- [tex]\(a + c > b\)[/tex]
- [tex]\(b + c > a\)[/tex]
Given:
- [tex]\(a = 29\)[/tex]
- [tex]\(b = 40\)[/tex]
Let's determine the constraints for the third side, [tex]\(c\)[/tex].
### Step-by-Step Calculation:
1. Determine the upper bound for [tex]\(c\)[/tex]:
According to the inequality theorem, [tex]\(a + b > c\)[/tex]. So,
[tex]\[ 29 + 40 > c \][/tex]
[tex]\[ 69 > c \][/tex]
This means that [tex]\(c\)[/tex] must be less than 69.
2. Determine the lower bound for [tex]\(c\)[/tex]:
We need to consider both [tex]\( |a - b| < c\)[/tex] and [tex]\(a \neq b \)[/tex], but since the specific constraint is more restrictive, it is more effective to use:
[tex]\[ |29 - 40| < c \][/tex]
This absolute difference is important:
[tex]\[ 11 < c \][/tex]
This means that [tex]\(c\)[/tex] must be greater than 11.
### Combining the Results:
From the above two points, the third side [tex]\(c\)[/tex] must satisfy:
[tex]\[ 11 < c < 69 \][/tex]
### Conclusion:
The correct range in which the length of the third side must lie is:
[tex]\[ 11 < x < 69 \][/tex]
Thus, option B and C are both correct.
[tex]\[
\boxed{11 < x < 69}
\][/tex]
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