Question

A triangle has two side lengths of 13.5 centimeters and 17.5 centimeters. What is a possible length for the third​ side? Explain why this is a possible length.
Question content area bottom
Part 1
A possible side length is
▼
4
4.5
3.5
3.75
centimeters.​ Then, the sum of the lengths of the two shortest sides is
  
enter your response here ​centimeters, which is
▼
less than
greater than
equal to
the longest side of
  
enter your response here centimeters.
​(Type whole numbers or​ decimals.)

Answer 1

To determine a possible length for the third side of a triangle with sides of lengths 13.5 centimeters and 17.5 centimeters, we can use the triangle inequality principle, which states:

1. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

For the given sides \( a = 13.5 \) cm and \( b = 17.5 \) cm, let's denote the third side as \( c \). According to the triangle inequality, we have the following three conditions:

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

Now let's apply these conditions:

1. From \( 13.5 + 17.5 > c \):
\[
31 > c \quad \text{or} \quad c < 31
\]

2. From \( 13.5 + c > 17.5 \):
\[
c > 4 \quad \text{or} \quad c > 4
\]

3. From \( 17.5 + c > 13.5 \):
\[
c > -4 \quad \text{(this inequality is always true since \( c \) must be positive)}
\]

Combining the valid inequalities, we get:
\[
4 < c < 31
\]
This means that any value for \( c \) that is greater than 4 cm and less than 31 cm is a valid length for the third side.

From the options provided (4, 4.5, 3.5, 3.75 cm), the only possible length for the third side is **4.5 cm**, as it meets the criteria \( 4 < c < 31 \).

### Answering the filling parts:

- A possible side length is **4.5** centimeters.
- The sum of the lengths of the two shortest sides (4.5 cm + 13.5 cm = 18 cm) is **18** centimeters.
- This sum (18 cm) is **greater than** the longest side of **17.5** centimeters.

So to complete the statements:
- A possible side length is **4.5** centimeters.
- The sum of the lengths of the two shortest sides is **18** centimeters, which is **greater than** the longest side of **17.5** centimeters.