To apply the Triangle Inequality Theorem, we need to understand that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's first convert all measurements to the same unit, either cm or mm. Since 1 cm = 10 mm, we can convert the sides:
- One side is \( 612 , \text{cm} \)
- The other side is \( 10 , \text{mm} = 1 , \text{cm} \)
Now, let's denote the third side as \( x , \text{cm} \). The Triangle Inequality Theorem gives us three conditions:
- \( 612 + 1 > x \)
- \( 612 + x > 1 \)
- \( 1 + x > 612 \)
Now, solving these inequalities:
- \( 613 > x \) or \( x < 613 \)
- \( 612 + x > 1 \) is always true, as \( x > -611 \) does not impose any restriction.
- \( 1 + x > 612 \) gives \( x > 611 \).
Combining the useful inequalities found, we have:
- \( 611 < x < 613 \)
The only integer that fits this range is \( 612 \).
Now looking at the answer choices:
- 6 cm: Not possible
- 16 cm: Not possible
- 18 cm: Not possible
- 8 cm: Not possible
No value is provided on your list that falls in the range of (611, 613). If the question allows for only integers, however, you might reconsider if \( 612 \) was among potential options (though you did not list it).
But based on your options, none are valid.
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