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.
A. equilateral
B. isosceles
C. scalene
D. acute

Answer 1

To classify the triangle by its sides given two side lengths of 1 cm and 13 cm, we first need to determine if a triangle can be formed with these sides along with the third side, which we'll denote as \(x\).

For a set of sides to form a triangle, they must satisfy the triangle inequality theorem, which states:
1. The sum of the lengths of any two sides must be greater than the length of the third side.

In this case, we have:
- Side 1: 1 cm
- Side 2: 13 cm
- Side 3: \(x\) cm (unknown for now)

Applying the triangle inequalities:
1. \(1 + 13 > x\) → \(14 > x\) → \(x < 14\)
2. \(1 + x > 13\) → \(x > 12\)
3. \(13 + x > 1\) → \(x > -12\) (This condition is always satisfied since \(x\) is a positive length)

From the inequalities:
- We found \(12 < x < 14\).

Since \(x\) must be a whole number, the only possible value for \(x\) is 13. Therefore, the side lengths are 1, 13, and 13.

Now, we can classify the triangle based on its side lengths:
- Two sides (13 cm and 13 cm) are equal, and one side (1 cm) is different.

This means the triangle is classified as **isosceles**.

Thus, the answer is **B. isosceles**.