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Answer :
To find the range for the length of the third side of a triangle when two sides are given, we use the triangle inequality theorem. According to this theorem, for any triangle:
1. The sum of the lengths of any two sides must be greater than the length of the third side.
Given sides:
- One side is 29 units.
- Another side is 40 units.
Let the third side be [tex]\( x \)[/tex].
To find the range for [tex]\( x \)[/tex], we have three inequalities based on the triangle inequality theorem:
1. [tex]\( x + 29 > 40 \)[/tex]
2. [tex]\( x + 40 > 29 \)[/tex]
3. [tex]\( 29 + 40 > x \)[/tex]
Let's solve these inequalities one by one:
1. [tex]\( x + 29 > 40 \)[/tex]
- Subtract 29 from both sides:
- [tex]\( x > 11 \)[/tex]
2. [tex]\( x + 40 > 29 \)[/tex]
- Subtract 40 from both sides:
- [tex]\( x > -11 \)[/tex]
Since the length of a side cannot be negative, this inequality is automatically satisfied as [tex]\( x \)[/tex] must be a positive number, and we've already determined [tex]\( x > 11 \)[/tex].
3. [tex]\( 29 + 40 > x \)[/tex]
- Simplify:
- [tex]\( 69 > x \)[/tex]
- This means [tex]\( x < 69 \)[/tex].
Combining the results from these inequalities, the third side [tex]\( x \)[/tex] must satisfy:
[tex]\[ 11 < x < 69 \][/tex]
Therefore, the correct option is:
C. [tex]\( 11 < x < 69 \)[/tex]
1. The sum of the lengths of any two sides must be greater than the length of the third side.
Given sides:
- One side is 29 units.
- Another side is 40 units.
Let the third side be [tex]\( x \)[/tex].
To find the range for [tex]\( x \)[/tex], we have three inequalities based on the triangle inequality theorem:
1. [tex]\( x + 29 > 40 \)[/tex]
2. [tex]\( x + 40 > 29 \)[/tex]
3. [tex]\( 29 + 40 > x \)[/tex]
Let's solve these inequalities one by one:
1. [tex]\( x + 29 > 40 \)[/tex]
- Subtract 29 from both sides:
- [tex]\( x > 11 \)[/tex]
2. [tex]\( x + 40 > 29 \)[/tex]
- Subtract 40 from both sides:
- [tex]\( x > -11 \)[/tex]
Since the length of a side cannot be negative, this inequality is automatically satisfied as [tex]\( x \)[/tex] must be a positive number, and we've already determined [tex]\( x > 11 \)[/tex].
3. [tex]\( 29 + 40 > x \)[/tex]
- Simplify:
- [tex]\( 69 > x \)[/tex]
- This means [tex]\( x < 69 \)[/tex].
Combining the results from these inequalities, the third side [tex]\( x \)[/tex] must satisfy:
[tex]\[ 11 < x < 69 \][/tex]
Therefore, the correct option is:
C. [tex]\( 11 < x < 69 \)[/tex]
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