Two sides of a triangles are 8 cm and 11 cm . The length of its third side lies between a cm b cm, find the values of a and b if a < 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.

Given:
- Side 1 (p) = 8 cm
- Side 2 (q) = 11 cm

Let the length of the third side be r.

### Step 1: Apply the Triangle Inequality Theorem
According to the triangle inequality theorem, we have the following inequalities:

1. \( p + q > r \)
2. \( p + r > q \)
3. \( q + r > p \)

### Step 2: Substitute the Known Values
Substituting the values of p and q into the inequalities:

1. \( 8 + 11 > r \) → \( 19 > r \) → \( r < 19 \)
2. \( 8 + r > 11 \) → \( r > 3 \)
3. \( 11 + r > 8 \) → \( r > -3 \) (This inequality is always satisfied since r is positive)

### Step 3: Combine the Inequalities
From the inequalities derived, we have:
- \( r > 3 \)
- \( r < 19 \)

This means the length of the third side (r) must lie between 3 cm and 19 cm.

### Step 4: Identify Values of a and b
From the inequalities, we can conclude:
- \( a = 3 \)
- \( b = 19 \)

Thus, the length of the third side lies between 3 cm and 19 cm.

### Final Answer
The values of a and b are:
- \( a = 3 \)
- \( b = 19 \)