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 equilateral equilateral acute acute scalene scalene isosceles isosceles

Answer 1

To determine if a triangle can exist with side lengths of 1 cm and 13 cm, we need to apply the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side.

Let's denote the sides as \(a = 1\) cm, \(b = 13\) cm, and let \(c\) be the length of the third side. According to the triangle inequality, the following must hold true:

1. \( a + b > c \) (1 + 13 > c)
2. \( a + c > b \) (1 + c > 13)
3. \( b + c > a \) (13 + c > 1)

From the first inequality, we find that:
- \(14 > c\), or \(c < 14\)

From the second inequality, we get:
- \(1 + c > 13\) leads to \(c > 12\)

From the third inequality:
- \(13 + c > 1\) is always true, as \(c\) is a positive length.

Thus from the inequalities \(c < 14\) and \(c > 12\), the only possible whole number for \(c\) that satisfies both inequalities is \(c = 13\).

Now we have the sides as:

- 1 cm, 13 cm, and 13 cm.

This means the triangle has two sides that are equal. Therefore, it is classified as an **isosceles triangle**.

So the answer is **isosceles**.