To find the possible lengths of the third side of a triangle when two sides are given, 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 the lengths of the two given sides be \( a = 10 \) mm and \( b = 7 \) mm. We can denote the length of the third side as \( c \).
According to the triangle inequality, we have the following conditions:
-
\( a + b > c \)
\( 10 + 7 > c \)
\( 17 > c \) or \( c < 17 \) -
\( a + c > b \)
\( 10 + c > 7 \)
\( c > 7 - 10 \)
\( c > -3 \) (which is always true for positive \( c \)) -
\( b + c > a \)
\( 7 + c > 10 \)
\( c > 10 - 7 \)
\( c > 3 \)
Combining these inequalities, we find that the length of the third side \( c \) must satisfy:
\( 3 < c < 17 \)
Now, let's evaluate the response options:
- 1 mm: does not satisfy \( c > 3 \)
- 2 mm: does not satisfy \( c > 3 \)
- 20 mm: does not satisfy \( c < 17 \)
- 5 mm: satisfies \( 3 < c < 17 \)
Thus, the only valid choice for the length of the third side from the given options is 5 mm.