A circle touch sides of a `Delta` ABC i.e AB, BC, AC at P,Q,R AP = 8 cm PB = 6 cm Find other sides if radius is 4 cm

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We have a triangle ABC with a circle inscribed in it, touching the sides AB, BC, and AC at points P, Q, and R respectively. We know the lengths AP = 8 cm, PB = 6 cm, and RC = 5 cm. We need to find the lengths of the sides AB, BC, and AC of triangle ABC.

### Step 2: Use the property of tangents
From the properties of tangents drawn from a point to a circle, we know that:
- The lengths of tangents drawn from a point outside the circle to the points of tangency are equal.

This gives us the following relationships:
- AP = AR
- BP = BQ
- CQ = CR

### Step 3: Calculate the lengths of the sides
1. **Calculate AB:**
- We have AP = 8 cm and PB = 6 cm.
- Therefore, AB = AP + PB = 8 cm + 6 cm = 14 cm.

2. **Calculate BC:**
- We have BP = 6 cm and CQ = CR (which we will find).
- We know that CR = 5 cm.
- Therefore, BC = BQ + QC = BP + CR = 6 cm + 5 cm = 11 cm.

3. **Calculate AC:**
- We have AR = AP (which we already know) and CR = 5 cm.
- Therefore, AC = AR + CR = 8 cm + 5 cm = 13 cm.

### Step 4: Summarize the results
- The lengths of the sides of triangle ABC are:
- AB = 14 cm
- BC = 11 cm
- AC = 13 cm

### Final Answer
The sides of triangle ABC are:
- AB = 14 cm
- BC = 11 cm
- AC = 13 cm