Answer :
To determine the range in which the length of the third side of a triangle must lie, we use the triangle inequality theorem. This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
Let's denote the sides of the triangle as [tex]\(a = 29\)[/tex], [tex]\(b = 40\)[/tex], and [tex]\(x\)[/tex], where [tex]\(x\)[/tex] is the length of the third side that we need to find.
According to the triangle inequality theorem, the following conditions must be met:
1. [tex]\(a + b > x\)[/tex]
2. [tex]\(a + x > b\)[/tex]
3. [tex]\(b + x > a\)[/tex]
Let's apply these conditions to find the range for [tex]\(x\)[/tex]:
1. [tex]\(29 + 40 > x\)[/tex], which simplifies to [tex]\(69 > x\)[/tex] or [tex]\(x < 69\)[/tex].
2. [tex]\(29 + x > 40\)[/tex], which simplifies to [tex]\(x > 40 - 29\)[/tex] or [tex]\(x > 11\)[/tex].
3. [tex]\(40 + x > 29\)[/tex] is always true as [tex]\(x\)[/tex] would need to be positive, and we already know [tex]\(x > 11\)[/tex].
Combining these results, the third side must satisfy:
[tex]\[ 11 < x < 69 \][/tex]
So, the correct range in which the length of the third side must lie is [tex]\(11 < x < 69\)[/tex].
The correct answer is:
C. [tex]\(11 < x < 69\)[/tex]
Let's denote the sides of the triangle as [tex]\(a = 29\)[/tex], [tex]\(b = 40\)[/tex], and [tex]\(x\)[/tex], where [tex]\(x\)[/tex] is the length of the third side that we need to find.
According to the triangle inequality theorem, the following conditions must be met:
1. [tex]\(a + b > x\)[/tex]
2. [tex]\(a + x > b\)[/tex]
3. [tex]\(b + x > a\)[/tex]
Let's apply these conditions to find the range for [tex]\(x\)[/tex]:
1. [tex]\(29 + 40 > x\)[/tex], which simplifies to [tex]\(69 > x\)[/tex] or [tex]\(x < 69\)[/tex].
2. [tex]\(29 + x > 40\)[/tex], which simplifies to [tex]\(x > 40 - 29\)[/tex] or [tex]\(x > 11\)[/tex].
3. [tex]\(40 + x > 29\)[/tex] is always true as [tex]\(x\)[/tex] would need to be positive, and we already know [tex]\(x > 11\)[/tex].
Combining these results, the third side must satisfy:
[tex]\[ 11 < x < 69 \][/tex]
So, the correct range in which the length of the third side must lie is [tex]\(11 < x < 69\)[/tex].
The correct answer is:
C. [tex]\(11 < x < 69\)[/tex]
