Answer :
Sure, we can determine the range of the third side of a triangle using 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. This leads to a few key inequalities based on a triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
For our triangle, we have two known sides: 29 units and 40 units. We need to find the possible range for the third side, which we'll call [tex]\(x\)[/tex].
Using the inequalities:
1. [tex]\(29 + 40 > x \Rightarrow x < 69\)[/tex]
2. [tex]\(29 + x > 40 \Rightarrow x > 11\)[/tex]
3. [tex]\(40 + x > 29 \Rightarrow x > -11\)[/tex] (This inequality isn't useful in this case since [tex]\(x > 11\)[/tex] is a tighter constraint.)
The useful inequalities are:
- [tex]\(x < 69\)[/tex] from the first inequality.
- [tex]\(x > 11\)[/tex] from the second inequality.
Therefore, by combining these, the length of the third side must lie in the range [tex]\(11 < x < 69\)[/tex].
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]
For our triangle, we have two known sides: 29 units and 40 units. We need to find the possible range for the third side, which we'll call [tex]\(x\)[/tex].
Using the inequalities:
1. [tex]\(29 + 40 > x \Rightarrow x < 69\)[/tex]
2. [tex]\(29 + x > 40 \Rightarrow x > 11\)[/tex]
3. [tex]\(40 + x > 29 \Rightarrow x > -11\)[/tex] (This inequality isn't useful in this case since [tex]\(x > 11\)[/tex] is a tighter constraint.)
The useful inequalities are:
- [tex]\(x < 69\)[/tex] from the first inequality.
- [tex]\(x > 11\)[/tex] from the second inequality.
Therefore, by combining these, the length of the third side must lie in the range [tex]\(11 < x < 69\)[/tex].
The correct answer is:
C. [tex]\(11 < x < 69\)[/tex]