Question

A) A triangle’s side lengths are whole numbers. If the measure of two of its sides are 1 cm and 13
cm, what is the measure of the third side? (1 point)
13 cm
12 cm
12.5 cm
1 cm

Answer 1

To determine the possible lengths of the third side of a triangle with the other two sides measuring 1 cm and 13 cm, we can use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let the lengths of the sides be \( a = 1 \) cm, \( b = 13 \) cm, and \( c \) be the unknown length of the third side. The triangle inequalities we need to check are:

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

Plugging in the known values:

1. \( 1 + 13 > c \) → \( 14 > c \) → \( c < 14 \)
2. \( 1 + c > 13 \) → \( c > 12 \)
3. \( 13 + c > 1 \) → This inequality will always be satisfied since \( c > 12 \).

From the inequalities \( c < 14 \) and \( c > 12 \), we can conclude that the third side \( c \) must be in the range \( 12 < c < 14 \). Since \( c \) is a whole number, the only possible value for \( c \) is 13.

However, that leads us to a side length equal to one of the existing sides, which is also acceptable in this context as a degenerate triangle.

Now we check the available options:

- 13 cm
- 12 cm
- 12.5 cm
- 1 cm

**The answer must fit the condition of \( 12 < c < 14 \).**

Thus, the only valid response among the given options is **13 cm**.