Question

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)%0D%0AResponses%0D%0A%0D%0A13 cm%0D%0A13 cm%0D%0A%0D%0A1 cm%0D%0A1 cm%0D%0A%0D%0A12.5 cm%0D%0A12.5 cm%0D%0A%0D%0A12 cm

Answer 1

To find the possible length of the third side of a triangle given the other two sides (1 cm and 13 cm), we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.

Let \( a = 1 \) cm, \( b = 13 \) cm, and \( c \) be the length of the third side.

According to the triangle inequality, we have the following conditions:

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

Now, let's simplify each condition:

1. From \( 1 + 13 > c \):
\[
14 > c \quad \Rightarrow \quad c < 14
\]

2. From \( 1 + c > 13 \):
\[
c > 12 \quad \Rightarrow \quad c > 12
\]

3. From \( 13 + c > 1 \):
\[
c > -12 \quad \text{(this condition doesn't restrict c because c must be positive)}
\]

Now we combine the valid inequalities:
- From \( c > 12 \) and \( c < 14 \), the only integer value that satisfies both conditions is:
\[
c = 13
\]

However, in a triangle, two sides cannot be equal to the third side; therefore, \( c \) cannot equal 13. The only integer length for the third side that satisfies all the inequalities is \( c = 12 \).

Thus, the measure of the third side can only be:

**12 cm**.