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

scalene
scalene

equilateral
equilateral

isosceles
isosceles

acute

Answer 1

To determine the classification of the triangle based on its side lengths (1 cm, 13 cm, and a third side), we should first consider the triangle inequality theorem, which states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Given the two sides of the triangle:
- One side = 1 cm
- Another side = 13 cm
- Let the third side be \(x\) cm.

The triangle inequality gives us three conditions:

1. \(1 + 13 > x\) ⟹ \(14 > x\) or \(x < 14\)
2. \(1 + x > 13\) ⟹ \(x > 12\)
3. \(13 + x > 1\) ⟹ This condition is always true since both 13 and \(x\) are positive.

Combining the first two inequalities:
- \(12 < x < 14\)

This implies that the only possible integer value for \(x\) can be \(13\), leading to the side lengths being \(1\), \(13\), and \(13\).

Now, classifying the triangle by its sides:
- Since we have one side of \(1\) cm and two sides of \(13\) cm, it can be classified as **isosceles** (it has two sides of equal length).

So the correct classification by its sides is: **isosceles**.