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Answer :
To determine the range in which the length of the third side of a triangle must lie, we can use the triangle inequality theorem. This theorem states that for any triangle with sides of lengths [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], the following inequalities must be satisfied:
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
Based on the given problem, we have two known sides of the triangle: one side is 29 units, and the other side is 40 units. Let's denote the length of the third side as [tex]\(x\)[/tex].
Using the triangle inequality theorem:
1. [tex]\(29 + 40 > x\)[/tex]
- This simplifies to [tex]\(x < 69\)[/tex].
2. [tex]\(29 + x > 40\)[/tex]
- This simplifies to [tex]\(x > 11\)[/tex].
3. [tex]\(40 + x > 29\)[/tex]
- This simplifies to [tex]\(x > -11\)[/tex], but since [tex]\(x\)[/tex] must be a positive length, this inequality does not affect our solution.
Considering all these conditions together, the only inequalities that matter are [tex]\(x < 69\)[/tex] and [tex]\(x > 11\)[/tex]. Thus, the length of the third side must satisfy:
[tex]\[ 11 < x < 69 \][/tex]
Therefore, the correct answer is:
C. [tex]\(11 < x < 69\)[/tex]
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
Based on the given problem, we have two known sides of the triangle: one side is 29 units, and the other side is 40 units. Let's denote the length of the third side as [tex]\(x\)[/tex].
Using the triangle inequality theorem:
1. [tex]\(29 + 40 > x\)[/tex]
- This simplifies to [tex]\(x < 69\)[/tex].
2. [tex]\(29 + x > 40\)[/tex]
- This simplifies to [tex]\(x > 11\)[/tex].
3. [tex]\(40 + x > 29\)[/tex]
- This simplifies to [tex]\(x > -11\)[/tex], but since [tex]\(x\)[/tex] must be a positive length, this inequality does not affect our solution.
Considering all these conditions together, the only inequalities that matter are [tex]\(x < 69\)[/tex] and [tex]\(x > 11\)[/tex]. Thus, the length of the third side must satisfy:
[tex]\[ 11 < x < 69 \][/tex]
Therefore, the correct answer is:
C. [tex]\(11 < x < 69\)[/tex]
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