A circle is inscribed in a `DeltaABC` touching AB, BC and AC at P, Q and R respectively. If AB = 10 cm, AR = 7 cm and CR = 5 cm, then find the length of BC.

Answer

To solve the problem, we need to find the length of side BC in triangle ABC, given the lengths of sides AB, AR, and CR. Let's break it down step by step.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- AB = 10 cm
- AR = 7 cm
- CR = 5 cm

2. **Define Variables for the Tangent Segments:**
- Let AP = AR = x (since the lengths from the vertex to the point of tangency are equal).
- Let BP = BQ = y (the length from B to the point of tangency on AC).
- Let CQ = CR = z (the length from C to the point of tangency on AB).

3. **Set Up the Equations:**
- From the triangle, we know:
- AB = AP + PB = x + y
- BC = BQ + QC = y + z
- AC = AR + RC = x + z

4. **Substitute the Known Values:**
- We know that AB = 10 cm, so:
\[ x + y = 10 \quad (1) \]
- We also know that CR = 5 cm, so:
\[ z = 5 \quad (2) \]
- We know that AR = 7 cm, so:
\[ x = 7 \quad (3) \]

5. **Substitute x into Equation (1):**
- From (3), substitute x into (1):
\[ 7 + y = 10 \]
- Solving for y:
\[ y = 10 - 7 = 3 \quad (4) \]

6. **Calculate BC:**
- Now, substitute y and z into the equation for BC:
\[ BC = y + z = 3 + 5 = 8 \text{ cm} \]

### Final Answer:
The length of BC is **8 cm**.