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)

Question

Responses

scalene
scalene

acute
acute

equilateral
equilateral

isosceles

Answer 1

To determine the classification of the triangle by its side lengths, we must apply the triangle inequality theorem. The triangle inequality states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following must be true:

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

In this case, the known side lengths are 1 cm and 13 cm. Let the third side be denoted as \(x\).

Now we can apply the triangle inequality:

1. \(1 + 13 > x \implies 14 > x\)
2. \(1 + x > 13 \implies x > 12\)
3. \(13 + x > 1\) (This condition is always satisfied since \(x\) is a positive whole number)

From the inequalities we derived:

• The first inequality gives us \(x < 14\).
• The second inequality gives us \(x > 12\).

Combining these, we find that \(x\) must be a whole number such that \(12 < x < 14\), which means \(x = 13\).

Therefore, the side lengths of the triangle are \(1\), \(13\), and \(13\), making it an isosceles triangle (since two sides are of equal length).

Thus, the correct classification of the triangle by its sides is:

isosceles