To determine whether the triangle with sides measuring 1 cm and 13 cm can form a valid triangle, we must apply 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 sides be \(a = 1\), \(b = 13\), and we'll call the third side \(c\).
According to the triangle inequality, we need to satisfy the following conditions:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Substituting \(1\) and \(13\) for \(a\) and \(b\):
- \(1 + 13 > c \implies 14 > c\) (Thus, \(c < 14\))
- \(1 + c > 13 \implies c > 12\)
- \(13 + c > 1 \implies c > -12\) (This condition is always satisfied for positive \(c\))
Now we combine the valid inequalities from the first two points:
- From \(c < 14\) and \(c > 12\), we conclude that the only possible integer value for \(c\) is \(13\).
Therefore, the side lengths of this triangle can only be \(1\), \(13\), and \(13\).
Since we have a triangle with two equal sides (1 cm and 13 cm), this makes the triangle isosceles.
So the triangle can be classified as:
Isosceles.
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