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 \leq x \leq 69 [/tex]

Answer :

To determine the range within 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 third side.

Let's consider the sides we already know:
- Side 1: 29 units
- Side 2: 40 units

Let the length of the third side be [tex]\( x \)[/tex].

According to the triangle inequality theorem, we have the following conditions:

1. [tex]\( x + 29 > 40 \)[/tex]
2. [tex]\( x + 40 > 29 \)[/tex]
3. [tex]\( 29 + 40 > x \)[/tex]

Now, let's solve these inequalities:

1. [tex]\( x + 29 > 40 \)[/tex]

Subtract 29 from both sides:
[tex]\[
x > 11
\][/tex]

2. [tex]\( x + 40 > 29 \)[/tex]

Subtract 40 from both sides:
[tex]\[
x > -11
\][/tex]
Since [tex]\( x \)[/tex] must be positive in a triangle, this condition doesn't affect our practical range because it's automatically satisfied by [tex]\( x > 11 \)[/tex].

3. [tex]\( 29 + 40 > x \)[/tex]

Simplify:
[tex]\[
69 > x \quad \text{or} \quad x < 69
\][/tex]

Combining the valid conditions, the length of the third side [tex]\( x \)[/tex] must satisfy:
[tex]\[
11 < x < 69
\][/tex]

Thus, the correct answer is:
C. [tex]\( 11 < x < 69 \)[/tex]

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