Answer :
To determine the range in which the length of the third side of a triangle must lie, we can apply the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Specifically, for a triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
In this problem, we have two sides given as 29 units and 40 units, and we need to determine the possible range for the third side, which we'll call [tex]\(x\)[/tex].
### Step 1: Apply the triangle inequality
1. [tex]\(29 + 40 > x\)[/tex]:
Simplifying gives [tex]\(69 > x\)[/tex], or [tex]\(x < 69\)[/tex].
2. [tex]\(29 + x > 40\)[/tex]:
Simplifying gives [tex]\(x > 11\)[/tex].
3. [tex]\(40 + x > 29\)[/tex]:
Simplifying gives [tex]\(x > -11\)[/tex], but since side lengths can't be negative, this condition is already covered by [tex]\(x > 11\)[/tex].
### Conclusion
Combining these inequalities, the range for the third side, [tex]\(x\)[/tex], is:
11 < x < 69
So the correct answer is option C: 11 < x < 69.
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
In this problem, we have two sides given as 29 units and 40 units, and we need to determine the possible range for the third side, which we'll call [tex]\(x\)[/tex].
### Step 1: Apply the triangle inequality
1. [tex]\(29 + 40 > x\)[/tex]:
Simplifying gives [tex]\(69 > x\)[/tex], or [tex]\(x < 69\)[/tex].
2. [tex]\(29 + x > 40\)[/tex]:
Simplifying gives [tex]\(x > 11\)[/tex].
3. [tex]\(40 + x > 29\)[/tex]:
Simplifying gives [tex]\(x > -11\)[/tex], but since side lengths can't be negative, this condition is already covered by [tex]\(x > 11\)[/tex].
### Conclusion
Combining these inequalities, the range for the third side, [tex]\(x\)[/tex], is:
11 < x < 69
So the correct answer is option C: 11 < x < 69.
