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Answer :
The problem involves finding the range of the third side of a triangle given the lengths of the other two sides. The triangle inequality theorem is applied, which states that the sum of any two sides of a triangle must be greater than the third side. This leads to the inequalities $29 + 40 > x$, $29 + x > 40$, and $40 + x > 29$. Solving these inequalities gives $x < 69$, $x > 11$, and $x > -11$. Combining these results, the range for the length of the third side is $11 < x < 69$. The final answer is $11 < x < 69$.
### Explanation
1. Applying the Triangle Inequality Theorem
Let $a$, $b$, and $c$ be the lengths of the sides of a triangle. The triangle inequality theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. In other words:
$a + b > c$
$a + c > b$
$b + c > a$
In this problem, we are given two sides of a triangle with lengths 29 and 40. Let $x$ be the length of the third side. We can apply the triangle inequality theorem to find the range of possible values for $x$.
2. Setting up Inequalities
We have the following inequalities:
$29 + 40 > x \Rightarrow 69 > x$
$29 + x > 40
\Rightarrow x > 40 - 29
\Rightarrow x > 11$
$40 + x > 29
\Rightarrow x > 29 - 40
\Rightarrow x > -11$
Since $x$ represents the length of a side, it must be positive. Therefore, $x > 0$. The inequality $x > -11$ is satisfied if $x > 11$, so we can ignore $x > -11$.
3. Finding the Range of Possible Values
Combining the inequalities $69 > x$ and $x > 11$, we get $11 < x < 69$. This means that the length of the third side must be greater than 11 and less than 69.
4. Final Answer
The correct answer is C. $11 < x < 69$.
### Examples
The triangle inequality is a fundamental concept in geometry and has practical applications in various fields. For example, in construction, when building a triangular structure, the lengths of the sides must satisfy the triangle inequality to ensure the structure is stable. Similarly, in navigation, the shortest distance between two points is a straight line, and any detour will always be longer, illustrating the triangle inequality.
### Explanation
1. Applying the Triangle Inequality Theorem
Let $a$, $b$, and $c$ be the lengths of the sides of a triangle. The triangle inequality theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. In other words:
$a + b > c$
$a + c > b$
$b + c > a$
In this problem, we are given two sides of a triangle with lengths 29 and 40. Let $x$ be the length of the third side. We can apply the triangle inequality theorem to find the range of possible values for $x$.
2. Setting up Inequalities
We have the following inequalities:
$29 + 40 > x \Rightarrow 69 > x$
$29 + x > 40
\Rightarrow x > 40 - 29
\Rightarrow x > 11$
$40 + x > 29
\Rightarrow x > 29 - 40
\Rightarrow x > -11$
Since $x$ represents the length of a side, it must be positive. Therefore, $x > 0$. The inequality $x > -11$ is satisfied if $x > 11$, so we can ignore $x > -11$.
3. Finding the Range of Possible Values
Combining the inequalities $69 > x$ and $x > 11$, we get $11 < x < 69$. This means that the length of the third side must be greater than 11 and less than 69.
4. Final Answer
The correct answer is C. $11 < x < 69$.
### Examples
The triangle inequality is a fundamental concept in geometry and has practical applications in various fields. For example, in construction, when building a triangular structure, the lengths of the sides must satisfy the triangle inequality to ensure the structure is stable. Similarly, in navigation, the shortest distance between two points is a straight line, and any detour will always be longer, illustrating the triangle inequality.
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