To determine the classification of the triangle with sides of lengths 1 cm and 13 cm, we must first ensure that a triangle can actually exist with these side lengths.
For any triangle with side lengths \( a \), \( b \), and \( c \), the triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the remaining side. Thus, for the sides of lengths 1 cm and 13 cm, we denote the third side as \( c \). The inequalities we need to consider are:
- \( 1 + 13 > c \) (which simplifies to \( c < 14 \))
- \( 1 + c > 13 \) (which simplifies to \( c > 12 \))
- \( 13 + c > 1 \) (which is always true for \( c \) being a positive length)
From the inequalities, we combine the relevant ones:
- From \( c < 14 \) and \( c > 12 \), we conclude that \( 12 < c < 14 \).
- The only whole number that satisfies this inequality is \( c = 13 \).
However, if \( c = 13 \), we have two sides of length 13 cm, and one side of length 1 cm. This configuration would mean we have at least two equal sides (13 cm each), making it an isosceles triangle.
Thus, the classification of the triangle by its sides can be determined as follows:
- It is not an equilateral triangle because all three sides are not equal.
- It is not a scalene triangle since it does have two equal sides.
- It is not an acute triangle since we cannot form a valid triangle.
Hence the classification is:
D. isosceles.
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