High School

Select the correct answer.

A triangle has one side of length 29 units and another of length 40 units. Determine the range in which the length of the third side must lie.

A. [tex] -11 \ < \ x \ < \ 69 [/tex]

B. [tex] 11 \leq x \leq 69 [/tex]

C. [tex] 11 \ < \ x \ < \ 69 [/tex]

D. [tex] 11 - x = 69 [/tex]

Answer :

To find the range for the third side of a triangle with sides measuring 29 units and 40 units, we can use the Triangle Inequality Theorem. This theorem states that for any 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]

In this problem, let's set [tex]\(a = 29\)[/tex] units and [tex]\(b = 40\)[/tex] units. The third side will be [tex]\(c\)[/tex], and we need to determine the possible lengths for [tex]\(c\)[/tex].

Let's analyze each inequality:

1. For [tex]\(a + b > c\)[/tex]:
[tex]\[
29 + 40 > c \quad \Rightarrow \quad 69 > c \quad \Rightarrow \quad c < 69
\][/tex]

2. For [tex]\(a + c > b\)[/tex]:
[tex]\[
29 + c > 40 \quad \Rightarrow \quad c > 40 - 29 \quad \Rightarrow \quad c > 11
\][/tex]

3. For [tex]\(b + c > a\)[/tex]:
[tex]\[
40 + c > 29 \quad \Rightarrow \quad c > -11 \quad \Rightarrow \quad \text{This inequality is always true since \(c > 11\) satisfies it.}
\][/tex]

Combining these inequalities, we get:
[tex]\(11 < c < 69\)[/tex]

Therefore, the length of the third side must be greater than 11 units and less than 69 units.

The correct answer is C. [tex]\(11 < x < 69\)[/tex].