To apply the Triangle Inequality Theorem, we need to ensure that the sum of the lengths of any two sides of the triangle is greater than the length of the third side. Here, we have two sides:
- First side \( a = 612 \) cm
- Second side \( b = 10 \) mm = \( 1 \) cm (since \( 10 \) mm = \( 0.1 \) cm)
Now, let's denote the third side as \( c \).
The Triangle Inequality Theorem gives us three inequalities to check:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Substituting the known values:
- \( 612 + 1 > c \) → \( 613 > c \) → \( c < 613 \)
- \( 612 + c > 1 \) → \( c > 1 - 612 \) → \( c > -611 \) (This doesn't provide useful information since \( c \) must be positive)
- \( 1 + c > 612 \) → \( c > 612 - 1 \) → \( c > 611 \)
Combining these inequalities:
- From \( c < 613 \)
- From \( c > 611 \)
This means \( 611 < c < 613 \).
Now, let's check which of the given options for \( c \) fits this range:
- 16 cm = 1600 mm (not valid)
- 6 cm = 600 mm (not valid)
- 18 cm = 1800 mm (not valid)
- 8 cm = 800 mm (not valid)
None of the options match the range \( 611 < c < 613 \) or are valid triangle side lengths based on the Triangle Inequality Theorem.
However, if you're looking for a possible measure, \( c \) must be between 611 cm and 613 cm, which means none of the options provided (6 cm, 8 cm, 16 cm, 18 cm) are valid. Hence, none of the options reflect a possible measure of the third side according to the parameters given.
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