To determine the possible lengths of the third side of a triangle with side lengths of 1 cm and 13 cm, we can apply the triangle inequality theorem, which states:
- The sum of the lengths of any two sides must be greater than the length of the third side.
- This must be true for all three pairs of sides.
Let the third side be \( x \). We have three inequalities to consider based on the sides 1 cm, 13 cm, and \( x \):
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\( 1 + 13 > x \) \ This simplifies to \( 14 > x \), or \( x < 14 \).
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\( 1 + x > 13 \) \ This simplifies to \( x > 12 \).
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\( 13 + x > 1 \) \ This simplifies to \( x > -12 \), which is always satisfied since \( x \) must be positive.
From the two inequalities \( x < 14 \) and \( x > 12 \):
- The only integer value that satisfies both conditions is \( x = 13 \).
Thus, the measure of the third side can only be \( 13 \) cm or perhaps less than 14 while being more than 12.
However, since it is also a side of the triangle, the only suitable integer value that fits all conditions is 13 cm.
So the answer is 13 cm.