Answer :
To determine the range in which the length of the third side of a triangle must lie, we 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 remaining side.
Let's consider the given sides of the triangle: one side is 29 units, and another is 40 units. The third side is the unknown, which we'll refer to as [tex]\( x \)[/tex].
The triangle inequality provides us with the following conditions:
1. The sum of the lengths of the first side and the third side must be greater than the length of the second side:
[tex]\[
x + 29 > 40
\][/tex]
By solving this inequality, we find:
[tex]\[
x > 11
\][/tex]
2. The sum of the lengths of the second side and the third side must be greater than the length of the first side:
[tex]\[
x + 40 > 29
\][/tex]
Solving this gives:
[tex]\[
x > -11
\][/tex]
Since lengths must be positive, this condition doesn't affect our previous result, [tex]\( x > 11 \)[/tex].
3. Finally, the sum of the lengths of the first side and the second side must be greater than the third side:
[tex]\[
29 + 40 > x
\][/tex]
Solving this inequality results in:
[tex]\[
x < 69
\][/tex]
Taking these conditions together, the third side [tex]\( x \)[/tex] must satisfy:
[tex]\[
11 < x < 69
\][/tex]
Thus, the correct range for the length of the third side [tex]\( x \)[/tex] is given by option C: [tex]\( 11 < x < 69 \)[/tex].
Let's consider the given sides of the triangle: one side is 29 units, and another is 40 units. The third side is the unknown, which we'll refer to as [tex]\( x \)[/tex].
The triangle inequality provides us with the following conditions:
1. The sum of the lengths of the first side and the third side must be greater than the length of the second side:
[tex]\[
x + 29 > 40
\][/tex]
By solving this inequality, we find:
[tex]\[
x > 11
\][/tex]
2. The sum of the lengths of the second side and the third side must be greater than the length of the first side:
[tex]\[
x + 40 > 29
\][/tex]
Solving this gives:
[tex]\[
x > -11
\][/tex]
Since lengths must be positive, this condition doesn't affect our previous result, [tex]\( x > 11 \)[/tex].
3. Finally, the sum of the lengths of the first side and the second side must be greater than the third side:
[tex]\[
29 + 40 > x
\][/tex]
Solving this inequality results in:
[tex]\[
x < 69
\][/tex]
Taking these conditions together, the third side [tex]\( x \)[/tex] must satisfy:
[tex]\[
11 < x < 69
\][/tex]
Thus, the correct range for the length of the third side [tex]\( x \)[/tex] is given by option C: [tex]\( 11 < x < 69 \)[/tex].
