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Answer :
To find the range for the third side of a triangle with sides measuring 29 units and 40 units, we can use the Triangle Inequality Theorem. This theorem states that for any triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], the following must be true:
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
In this problem, let's set [tex]\(a = 29\)[/tex] units and [tex]\(b = 40\)[/tex] units. The third side will be [tex]\(c\)[/tex], and we need to determine the possible lengths for [tex]\(c\)[/tex].
Let's analyze each inequality:
1. For [tex]\(a + b > c\)[/tex]:
[tex]\[
29 + 40 > c \quad \Rightarrow \quad 69 > c \quad \Rightarrow \quad c < 69
\][/tex]
2. For [tex]\(a + c > b\)[/tex]:
[tex]\[
29 + c > 40 \quad \Rightarrow \quad c > 40 - 29 \quad \Rightarrow \quad c > 11
\][/tex]
3. For [tex]\(b + c > a\)[/tex]:
[tex]\[
40 + c > 29 \quad \Rightarrow \quad c > -11 \quad \Rightarrow \quad \text{This inequality is always true since \(c > 11\) satisfies it.}
\][/tex]
Combining these inequalities, we get:
[tex]\(11 < c < 69\)[/tex]
Therefore, the length of the third side must be greater than 11 units and less than 69 units.
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]
In this problem, let's set [tex]\(a = 29\)[/tex] units and [tex]\(b = 40\)[/tex] units. The third side will be [tex]\(c\)[/tex], and we need to determine the possible lengths for [tex]\(c\)[/tex].
Let's analyze each inequality:
1. For [tex]\(a + b > c\)[/tex]:
[tex]\[
29 + 40 > c \quad \Rightarrow \quad 69 > c \quad \Rightarrow \quad c < 69
\][/tex]
2. For [tex]\(a + c > b\)[/tex]:
[tex]\[
29 + c > 40 \quad \Rightarrow \quad c > 40 - 29 \quad \Rightarrow \quad c > 11
\][/tex]
3. For [tex]\(b + c > a\)[/tex]:
[tex]\[
40 + c > 29 \quad \Rightarrow \quad c > -11 \quad \Rightarrow \quad \text{This inequality is always true since \(c > 11\) satisfies it.}
\][/tex]
Combining these inequalities, we get:
[tex]\(11 < c < 69\)[/tex]
Therefore, the length of the third side must be greater than 11 units and less than 69 units.
The correct answer is C. [tex]\(11 < x < 69\)[/tex].
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