To determine the range in which the length of the third side of a triangle must lie, given two sides of lengths 29 units and 40 units, we apply 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 remaining side.
Here’s the detailed, step-by-step process:
1. Define the sides:
- Let side1 [tex]\( = 29 \)[/tex] units.
- Let side2 [tex]\( = 40 \)[/tex] units.
- Let side3 be the length of the third side, which we need to find.
2. Apply the triangle inequality theorem:
- According to the theorem, the following inequalities must be satisfied:
1. [tex]\( \text{side1} + \text{side2} > \text{side3} \)[/tex]
2. [tex]\( \text{side1} + \text{side3} > \text{side2} \)[/tex]
3. [tex]\( \text{side2} + \text{side3} > \text{side1} \)[/tex]
3. Substituting the given values into the inequalities:
1. [tex]\( 29 + 40 > \text{side3} \)[/tex]
2. [tex]\( 29 + \text{side3} > 40 \)[/tex]
3. [tex]\( 40 + \text{side3} > 29 \)[/tex]
4. Simplify the inequalities:
1. [tex]\( 69 > \text{side3} \)[/tex] or [tex]\( \text{side3} < 69 \)[/tex]
2. [tex]\( \text{side3} > 40 - 29 \)[/tex] or [tex]\( \text{side3} > 11 \)[/tex]
3. [tex]\( 40 + \text{side3} > 29 \)[/tex] or [tex]\( \text{side3} > -11 \)[/tex] (This inequality is always true for positive lengths and doesn't restrict side3 further.)
5. Combine the simplified inequalities:
- [tex]\( 11 < \text{side3} < 69 \)[/tex]
Therefore, the third side must be greater than 11 units and less than 69 units for it to form a valid triangle with the given sides.
6. Select the correct answer:
- The correct range for the third side is [tex]\( 11 < x < 69 \)[/tex].
So, the correct answer is: C. \[tex]$11 < x < 69\$[/tex]
