Answer :
To find the range in which the length of the third side must lie, we use the triangle inequality theorem. This theorem states that, for a triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], the following must be true:
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
Given that two sides of the triangle are 29 units and 40 units, let's denote the third side as [tex]\(x\)[/tex].
1. First, the third side must be less than the sum of the other two sides:
[tex]\[
x < 29 + 40 = 69
\][/tex]
2. Second, the third side must be greater than the absolute difference of the other two sides:
[tex]\[
x > |29 - 40| = 11
\][/tex]
Therefore, the length of the third side [tex]\(x\)[/tex] must satisfy:
[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]
Given that two sides of the triangle are 29 units and 40 units, let's denote the third side as [tex]\(x\)[/tex].
1. First, the third side must be less than the sum of the other two sides:
[tex]\[
x < 29 + 40 = 69
\][/tex]
2. Second, the third side must be greater than the absolute difference of the other two sides:
[tex]\[
x > |29 - 40| = 11
\][/tex]
Therefore, the length of the third side [tex]\(x\)[/tex] must satisfy:
[tex]\[
11 < x < 69
\][/tex]
The correct answer is C. [tex]\(11 < x < 69\)[/tex].
