A circle touches the side BC of a `Delta`ABC at P and also touches AB and AC produced at Q and R, respectively. If the perimeter of `Delta`ABC = 26.4 cm, then the length of AQ is:

Options

A. `8.8` cm
B. `13.2` cm
C. `15.4` cm
D. `17.6` cm

Answer

To solve the problem, we need to find the length of \( AQ \) given that a circle touches the sides of triangle \( ABC \) and the perimeter of the triangle is \( 26.4 \) cm.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The circle touches side \( BC \) at point \( P \).
- It also touches the extensions of sides \( AB \) and \( AC \) at points \( Q \) and \( R \), respectively.
- We need to find the length of \( AQ \).

2. **Using the Tangent Property**:
- From point \( A \), the lengths of the tangents drawn to the circle are equal. Thus:
- \( AQ = AP \) (tangent from \( A \) to the circle at point \( Q \))
- \( BP = BQ \) (tangent from \( B \) to the circle at point \( P \))
- \( CP = CR \) (tangent from \( C \) to the circle at point \( R \))

3. **Setting Up the Variables**:
- Let \( AQ = x \).
- Then \( AP = x \) (since \( AQ = AP \)).
- Let \( BP = y \) and \( CP = z \).
- Therefore, we can express the sides of the triangle as:
- \( AB = AQ + BP = x + y \)
- \( BC = BP + CP = y + z \)
- \( CA = CP + AQ = z + x \)

4. **Writing the Perimeter Equation**:
- The perimeter of triangle \( ABC \) is given as:
\[ AB + BC + CA = 26.4 \]
- Substituting the expressions for the sides:
\[ (x + y) + (y + z) + (z + x) = 26.4 \]
- Simplifying this gives:
\[ 2x + 2y + 2z = 26.4 \]
- Dividing the entire equation by 2:
\[ x + y + z = 13.2 \]

5. **Finding \( AQ \)**:
- Since \( AQ = x \), we can express \( y \) and \( z \) in terms of \( x \):
\[ y + z = 13.2 - x \]
- However, we need to find \( AQ \) directly. Since \( AQ \) is equal to \( AP \) and \( AP \) is one of the tangents, we can use the fact that the total of the tangents from point \( A \) to the circle is equal to half the perimeter:
\[ AQ = \frac{26.4}{2} = 13.2 \text{ cm} \]

6. **Final Calculation**:
- Since \( AQ \) is half of the perimeter, we find that:
\[ AQ = 13.2 \text{ cm} \]

### Conclusion:
The length of \( AQ \) is \( 13.2 \) cm.