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Answer :
To determine the range in which the length of the third side of the triangle must lie, we use the triangle inequality theorem. This theorem states that 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]
Given the two sides of the triangle are 29 units and 40 units, we'll denote the third side as [tex]\(x\)[/tex]. Let's apply the triangle inequality theorem to find the possible range for [tex]\(x\)[/tex].
1. Using [tex]\(a + b > c\)[/tex]:
This becomes [tex]\(29 + 40 > x\)[/tex].
Thus, [tex]\(x < 69\)[/tex].
2. Using [tex]\(a + c > b\)[/tex]:
This becomes [tex]\(29 + x > 40\)[/tex].
Solving for [tex]\(x\)[/tex], we get [tex]\(x > 40 - 29\)[/tex].
Therefore, [tex]\(x > 11\)[/tex].
3. Using [tex]\(b + c > a\)[/tex]:
This becomes [tex]\(40 + x > 29\)[/tex].
Solving for [tex]\(x\)[/tex], we get [tex]\(x > 29 - 40\)[/tex].
Since the subtraction results in a negative number, this condition, when solved, simply repeats the same inequality from step 2, which is [tex]\(x > 11\)[/tex].
Combining these conditions, we get that the range for the third side [tex]\(x\)[/tex] must be:
[tex]\[ 11 < x < 69 \][/tex]
So, the correct answer is option C: [tex]\(11 < x < 69\)[/tex].
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
Given the two sides of the triangle are 29 units and 40 units, we'll denote the third side as [tex]\(x\)[/tex]. Let's apply the triangle inequality theorem to find the possible range for [tex]\(x\)[/tex].
1. Using [tex]\(a + b > c\)[/tex]:
This becomes [tex]\(29 + 40 > x\)[/tex].
Thus, [tex]\(x < 69\)[/tex].
2. Using [tex]\(a + c > b\)[/tex]:
This becomes [tex]\(29 + x > 40\)[/tex].
Solving for [tex]\(x\)[/tex], we get [tex]\(x > 40 - 29\)[/tex].
Therefore, [tex]\(x > 11\)[/tex].
3. Using [tex]\(b + c > a\)[/tex]:
This becomes [tex]\(40 + x > 29\)[/tex].
Solving for [tex]\(x\)[/tex], we get [tex]\(x > 29 - 40\)[/tex].
Since the subtraction results in a negative number, this condition, when solved, simply repeats the same inequality from step 2, which is [tex]\(x > 11\)[/tex].
Combining these conditions, we get that the range for the third side [tex]\(x\)[/tex] must be:
[tex]\[ 11 < x < 69 \][/tex]
So, the correct answer is option C: [tex]\(11 < x < 69\)[/tex].
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