To determine the classification of the triangle by its sides, we need to 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's denote the side lengths as follows:
- Side 1: 1 cm
- Side 2: 13 cm
- Side 3: \( x \) (the length of the third side, which is a whole number)
According to the triangle inequality:
- \( 1 + 13 > x \) → \( 14 > x \)
- \( 1 + x > 13 \) → \( x > 12 \)
- \( 13 + x > 1 \) → This inequality is always true for positive \( x \).
From the inequalities we derived:
- From the first inequality, \( x < 14 \)
- From the second inequality, \( x > 12 \)
Thus, the only integer value that satisfies both inequalities is:
- \( x = 13 \)
Now we have the side lengths: 1 cm, 13 cm, and 13 cm. The classification by its sides is as follows:
- The triangle has two sides of equal length (13 cm) and one side of different length (1 cm).
Therefore, the triangle is classified as isosceles.
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