To determine the possible lengths of the third side of a triangle given the lengths of the first two sides, we can apply the Triangle Inequality Theorem, which states that for any triangle with sides of lengths \( a \), \( b \), and \( c \):
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
In this case, let's call the two known sides \( a = 4 \) and \( b = 6 \), and let \( c \) be the length of the third side.
Applying the Triangle Inequality Theorem:
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From \( a + b > c \): \[ 4 + 6 > c \implies 10 > c \implies c < 10 \]
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From \( a + c > b \): \[ 4 + c > 6 \implies c > 2 \]
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From \( b + c > a \): \[ 6 + c > 4 \implies c > -2 \text{ (this is always true since c is positive)} \]
From the inequalities we deduced, we can conclude that: \[ 2 < c < 10 \]
Now, let’s evaluate each option to see which values fall within the range \( 2 < c < 10 \):
- 9 (valid, since \( 2 < 9 < 10 \))
- 2 (not valid, since \( 2 \) is not greater than \( 2 \))
- 11 (not valid, since \( 11 \) is not less than \( 10 \))
- 1 (not valid, since \( 1 < 2 \))
Thus, the only valid option for the length of the third side \( c \) satisfying the Triangle Inequality Theorem is 9.