If a circle with radius 4 cm incircles the triangle abc and touches side ab ,bc,and ca respectively at points p q and r then find sides ab bc and ca if sides ap ,bq and cr are consecutive integers

Given the problem, we have a triangle \(ABC\) with an incircle that touches sides \(AB\), \(BC\), and \(CA\) at points \(P\), \(Q\), and \(R\), respectively. The radius of the incircle is 4 cm, and the lengths of segments \(AP\), \(BQ\), and \(CR\) are consecutive integers.

### Steps to Solve:

1. **Understanding the Problem**:
   - The incircle of the triangle touches the sides at points \(P\), \(Q\), and \(R\).
   - Let the sides of the triangle be \(AB = c\), \(BC = a\), and \(CA = b\).
   - Denote \(AP = s - a\), \(BQ = s - b\), and \(CR = s - c\), where \(s\) is the semi-perimeter of the triangle.

2. **Let the consecutive integers be**:
   - \(AP = x\)
   - \(BQ = x+1\)
   - \(CR = x+2\)

3. **Relationship between segments**:
   - Since the triangle is circumscribed, the lengths of the triangle's sides are given by:
     \[
     AB = c = BQ + CR = (x + 1) + (x + 2) = 2x + 3
     \]
     \[
     BC = a = CR + AP = (x + 2) + x = 2x + 2
     \]
     \[
     CA = b = AP + BQ = x + (x + 1) = 2x + 1
     \]

4. **Using the inradius formula**:
   - The area \(A\) of a triangle can be expressed using the formula:
     \[
     A = r \times s
     \]
     where \(r\) is the inradius and \(s\) is the semi-perimeter.
   - The semi-perimeter \(s\) is:
     \[
     s = \frac{a + b + c}{2} = \frac{(2x + 2) + (2x + 1) + (2x + 3)}{2} = 3x + 3
     \]
   - The area can also be calculated using Heron's formula:
     \[
     A = \sqrt{s(s-a)(s-b)(s-c)}
     \]
   - Equating the two expressions for the area and solving for \(x\) will yield the side lengths.

Let's calculate the value of \(x\) next.The solutions for \(x\) are \(-1\) and \(6\). Since \(x\) represents the length of a side segment and lengths cannot be negative, we discard \(-1\) and accept \(x = 6\).

### Now, using \(x = 6\):
- \(AP = x = 6\)
- \(BQ = x + 1 = 7\)
- \(CR = x + 2 = 8\)

### Therefore, the side lengths of the triangle are:
- \(AB = c = 2x + 3 = 2(6) + 3 = 15\)
- \(BC = a = 2x + 2 = 2(6) + 2 = 14\)
- \(CA = b = 2x + 1 = 2(6) + 1 = 13\)

Thus, the sides of the triangle are \(AB = 15\) cm, \(BC = 14\) cm, and \(CA = 13\) cm.

Would you like further details or have any questions?

### Related Questions:
1. How do we calculate the area of a triangle using Heron's formula?
2. What are other methods to find the inradius of a triangle?
3. How does the semi-perimeter relate to the sides of a triangle?
4. Can you explain the steps of deriving Heron's formula?
5. How would the solution change if the radius was different?

### Tip:
Always check for the feasibility of the solution in geometry problems, ensuring lengths are positive and within logical bounds.

Explore the solution to a geometry problem involving a triangle with an incircle and consecutive integers touching sides. Discover how to find the side lengths of triangle ABC given specific conditions.

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