in a figure a triangle abc is drawn to circumscribe a circle of radius 3cm such that the segments BD and DC into which BC is divided by the point f contact D are of lengths 6cm and 9cm respectively. If the area of abc = 54cm2 then find the lengths of sides AB and AC

To solve this problem, we can use some important properties of a triangle that circumscribes a circle.

### Key Properties:
1. In a triangle \(ABC\) circumscribing a circle, the sum of the lengths of the two tangents drawn from an external point to the points of contact with the circle are equal. 
   
   Let:
   - \(AB = c\), \(BC = a\), and \(CA = b\).
   - The point of tangency on \(BC\) is \(D\), on \(CA\) is \(E\), and on \(AB\) is \(F\).
   - The tangents from \(B\) and \(C\) to the circle are such that \(BD = 6 \, \text{cm}\) and \(DC = 9 \, \text{cm}\), so:
     \[
     a = BD + DC = 6 + 9 = 15 \, \text{cm}.
     \]
   
2. If the circle has a radius \(r\) and the area of \(\triangle ABC\) is \(A\), then the semi-perimeter \(s\) of \(\triangle ABC\) is related to \(A\) and \(r\) by:
   \[
   A = r \times s.
   \]
   
### Step-by-Step Solution:
#### 1. Calculate the Semi-perimeter \(s\):
Given:
- Radius \(r = 3 \, \text{cm}\)
- Area \(A = 54 \, \text{cm}^2\)

Using \(A = r \times s\):
\[
54 = 3 \times s
\]
\[
s = \frac{54}{3} = 18 \, \text{cm}.
\]

#### 2. Set Up Equations for \(AB\) and \(AC\):
Since \(s = \frac{a + b + c}{2}\), we have:
\[
18 = \frac{15 + b + c}{2}
\]
Multiplying both sides by 2:
\[
36 = 15 + b + c
\]
\[
b + c = 21.
\]

Let:
- \(AB = c\)
- \(AC = b\)

Also, from the tangents' property:
- \(AF = AE\)
- \(BF = BD = 6 \, \text{cm}\)
- \(CE = DC = 9 \, \text{cm}\)

Thus:
\[
c = AF + FB = AE + 6
\]
\[
b = AE + CE = AE + 9
\]

Since \(b + c = 21\), and substituting the values:


In a figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 6 cm and 9 cm, respectively. If the area of triangle ABC is 54 cm², find the lengths of sides AB and AC.

Explore how to find the lengths of sides AB and AC in a triangle ABC that circumscribes a circle with radius 3 cm. With given segments BD = 6 cm, DC = 9 cm, and an area of 54 cm², this solution applies properties of circumscribed circles and tangents to solve for AB and AC. Ideal for students in Grades 10-12.

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