To determine the possible length of the third side of a triangle when two sides are given, we can use the triangle inequality theorem. According to the theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Given the sides \( a = 10 \) mm and \( b = 7 \) mm, let's denote the third side as \( c \). The triangle inequalities can be expressed as:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Substituting the values we have:
- \( 10 + 7 > c \) → \( 17 > c \) (or \( c < 17 \))
- \( 10 + c > 7 \) → \( c > -3 \) (this inequality is always true for positive values)
- \( 7 + c > 10 \) → \( c > 3 \)
Combining the relevant inequalities, we find:
\[ 3 < c < 17 \]
Now, we can look at the given options: 2 mm, 1 mm, 20 mm, and 5 mm.
- 2 mm: Does not satisfy \( c > 3 \).
- 1 mm: Does not satisfy \( c > 3 \).
- 20 mm: Does not satisfy \( c < 17 \).
- 5 mm: Satisfies \( 3 < c < 17 \).
The only valid option is 5 mm. Thus, the length of the third side must be 5 mm.
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