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

scalene
scalene

acute
acute

equilateral

Answer 1

To determine the classification of the triangle based on its sides when two sides are 1 cm and 13 cm, we first need to apply the triangle inequality theorem. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's denote the sides of the triangle as:
- Side a = 1 cm
- Side b = 13 cm
- Side c = (unknown, but must also be a whole number)

We need to find a value for c such that all triangle inequality conditions are satisfied:

1. \( a + b > c \)
- \( 1 + 13 > c \)
- \( 14 > c \) → \( c < 14 \)

2. \( a + c > b \)
- \( 1 + c > 13 \)
- \( c > 12 \)

3. \( b + c > a \)
- \( 13 + c > 1 \)
- This condition is always true for any positive c.

From the inequalities:
- From \( c < 14 \) and \( c > 12 \), we see that the only whole number value for c that satisfies both conditions is 13 cm.

Thus, the three side lengths of the triangle are:
- a = 1 cm
- b = 13 cm
- c = 13 cm

Now, we can classify the triangle by its sides:
- Since two sides (b and c) are equal, the triangle is **isosceles**.

The correct response is:
**isosceles**