To determine the possible lengths of the third side of a triangle when two sides are known, we can use 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 third side.
Given sides of lengths \( a = 6 \) cm and \( b = 10 \) cm, we can denote the length of the third side as \( c \). The triangle inequalities can be expressed as:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Substituting the known values into these inequalities:
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\( 6 + 10 > c \)
- \( 16 > c \)
- \( c < 16 \)
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\( 6 + c > 10 \)
- \( c > 4 \)
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\( 10 + c > 6 \)
- This inequality is always true for positive \( c \).
Putting together these results, we find the range for \( c \) is:
\[ 4 < c < 16 \]
Now, we can evaluate the options provided:
- Option #1: 3 cm (not valid since \( c > 4 \))
- Option #2: 9 cm (valid since \( 4 < 9 < 16 \))
- Option #3: 16 cm (not valid since \( c < 16 \))
- Option #4: 20 cm (not valid since \( c < 16 \))
Therefore, the only valid option for the length of the third side is:
Option #2: 9 cm.
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