College

A triangle has one side of length 29 units and another of length 40 units. Determine the range in which the length of the third side must lie.

A. [tex] -11 < x < 69 [/tex]

B. [tex] 11 < x < 69 [/tex]

C. [tex] 11 < x < 69 [/tex]

D. [tex] -11 = x = 69 [/tex]

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]