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Answer :
To determine the range for the length of the third side of a triangle, we can use 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 third side. Let's apply this to the given triangle:
1. Label the sides:
- Let the lengths of the sides be labeled as [tex]\( a = 29 \)[/tex], [tex]\( b = 40 \)[/tex], and [tex]\( x \)[/tex] for the unknown side.
2. Apply the triangle inequality theorem:
We need to check three inequalities:
- [tex]\( x + a > b \)[/tex]
- [tex]\( x + b > a \)[/tex]
- [tex]\( a + b > x \)[/tex]
3. Simplify these inequalities:
First Inequality:
- [tex]\( x + 29 > 40 \)[/tex]
- Subtract 29 from both sides: [tex]\( x > 40 - 29 \)[/tex]
- This gives: [tex]\( x > 11 \)[/tex]
Second Inequality:
- [tex]\( x + 40 > 29 \)[/tex]
- Subtract 40 from both sides: [tex]\( x > 29 - 40 \)[/tex]
- Since [tex]\( 29 - 40 = -11 \)[/tex], this gives: [tex]\( x > -11 \)[/tex]
- However, [tex]\( x > 11 \)[/tex] from the first inequality is stricter than [tex]\( x > -11 \)[/tex], so we use [tex]\( x > 11 \)[/tex].
Third Inequality:
- [tex]\( 29 + 40 > x \)[/tex]
- Simplify: [tex]\( 69 > x \)[/tex]
- This gives: [tex]\( x < 69 \)[/tex]
4. Combine the inequalities:
- The two important inequalities we derived are [tex]\( x > 11 \)[/tex] and [tex]\( x < 69 \)[/tex].
5. Conclusion:
- Therefore, the range for the length of the third side [tex]\( x \)[/tex] is [tex]\( 11 < x < 69 \)[/tex].
Thus, the correct answer is option C: [tex]\( 11 < x < 69 \)[/tex].
1. Label the sides:
- Let the lengths of the sides be labeled as [tex]\( a = 29 \)[/tex], [tex]\( b = 40 \)[/tex], and [tex]\( x \)[/tex] for the unknown side.
2. Apply the triangle inequality theorem:
We need to check three inequalities:
- [tex]\( x + a > b \)[/tex]
- [tex]\( x + b > a \)[/tex]
- [tex]\( a + b > x \)[/tex]
3. Simplify these inequalities:
First Inequality:
- [tex]\( x + 29 > 40 \)[/tex]
- Subtract 29 from both sides: [tex]\( x > 40 - 29 \)[/tex]
- This gives: [tex]\( x > 11 \)[/tex]
Second Inequality:
- [tex]\( x + 40 > 29 \)[/tex]
- Subtract 40 from both sides: [tex]\( x > 29 - 40 \)[/tex]
- Since [tex]\( 29 - 40 = -11 \)[/tex], this gives: [tex]\( x > -11 \)[/tex]
- However, [tex]\( x > 11 \)[/tex] from the first inequality is stricter than [tex]\( x > -11 \)[/tex], so we use [tex]\( x > 11 \)[/tex].
Third Inequality:
- [tex]\( 29 + 40 > x \)[/tex]
- Simplify: [tex]\( 69 > x \)[/tex]
- This gives: [tex]\( x < 69 \)[/tex]
4. Combine the inequalities:
- The two important inequalities we derived are [tex]\( x > 11 \)[/tex] and [tex]\( x < 69 \)[/tex].
5. Conclusion:
- Therefore, the range for the length of the third side [tex]\( x \)[/tex] is [tex]\( 11 < x < 69 \)[/tex].
Thus, the correct answer is option C: [tex]\( 11 < x < 69 \)[/tex].
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