To find the possible length of the third side of a triangle with side lengths of 1 cm and 13 cm, we can use the triangle inequality theorem:
- The sum of the lengths of any two sides must be greater than the length of the remaining side.
Given sides \( a = 1 \) cm and \( b = 13 \) cm, we can denote the third side as \( c \). The triangle inequality gives us the following conditions:
-
\( a + b > c \)
- \( 1 + 13 > c \)
- \( 14 > c \)
- Thus, \( c < 14 \)
-
\( a + c > b \)
- \( 1 + c > 13 \)
- \( c > 12 \)
-
\( b + c > a \)
- \( 13 + c > 1 \)
- This condition is always satisfied because \( c \) is a positive length.
From the inequalities derived, we have:
- \( c < 14 \)
- \( c > 12 \)
Combining these, we find that the only integer value that satisfies \( 12 < c < 14 \) is:
- \( c = 13 \) cm
Therefore, the measure of the third side must be 13 cm.