Question

A triangle’s side lengths are whole numbers. If the measure of two of its sides are 1 cm and 13 cm, classify the triangle by its sides.(1 point)
Responses

isosceles
isosceles

equilateral
equilateral

acute
acute

scalene

Answer 1

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:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

Substituting \(1\) and \(13\) for \(a\) and \(b\):

1. \(1 + 13 > c \implies 14 > c\) (Thus, \(c < 14\))
2. \(1 + c > 13 \implies c > 12\)
3. \(13 + c > 1 \implies c > -12\) (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**.