If two sides of a triangle are 8 cm and 13 cm, then the length of the third side is between a cm and b cm. Find the values of a and b such that a is less than b.

Answer

To find the possible lengths of the third side of a triangle when two sides are given, we can use the triangle inequality theorem. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side, and the difference between the lengths of any two sides must be less than the length of the third side.

### Step-by-step Solution:

1. **Identify the lengths of the two sides**:
Let the lengths of the two sides be \( a_1 = 8 \) cm and \( a_2 = 13 \) cm.

2. **Apply the triangle inequality theorem**:
- The sum of the two sides must be greater than the third side:
\[ a_1 + a_2 > p \]
Substituting the values:
\[ 8 + 13 > p \implies 21 > p \implies p < 21 \]
- The difference of the two sides must be less than the third side:
\[ |a_2 - a_1| < p \]
Substituting the values:
\[ 13 - 8 < p \implies 5 < p \]

3. **Combine the inequalities**:
From the two inequalities we derived:
\[ 5 < p < 21 \]

4. **Identify the values of \( a \) and \( b \)**:
Here, \( a \) is the lower bound and \( b \) is the upper bound of the length of the third side. Thus:
\[ a = 5 \text{ cm} \quad \text{and} \quad b = 21 \text{ cm} \]

5. **Final answer**:
The values of \( a \) and \( b \) such that \( a < b \) are:
\[ a = 5 \text{ cm}, \quad b = 21 \text{ cm} \]