To determine which lengths could be the length of the third side of a triangle given the other two sides, 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.
Let’s denote the sides of the triangle as follows:
- \( a = 14 \) inches (first side)
- \( b = 6 \) inches (second side)
- \( c \) (third side, which we need to find)
According to the triangle inequality theorem, the following must be true:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Applying these inequalities:
- \( 14 + 6 > c \) → \( 20 > c \) → \( c < 20 \)
- \( 14 + c > 6 \) → \( c > -8 \) (this inequality does not impose a restriction since \( c \) must be positive)
- \( 6 + c > 14 \) → \( c > 8 \)
Now we combine the conditions:
- From the first inequality, we find \( c < 20 \).
- From the third inequality, we find \( c > 8 \).
So, the length of the third side \( c \) must satisfy:
\[ 8 < c < 20 \]
Now examining the given options:
- 7 inches: Does not satisfy \( c > 8 \) (not possible)
- 20 inches: Does not satisfy \( c < 20 \) (not possible)
- 15 inches: Satisfies \( 8 < 15 < 20 \) (possible)
- 3 inches: Does not satisfy \( c > 8 \) (not possible)
The only option that could be the length of the third side is 15 inches.
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