A triangle $ABC$ is drawn to circumscribe a circle of radius $4\text{ cm}$ such that the segments $BD$ and $DC$ into which $BC$ is divided by the point of contact $D$ are of lengths $8\text{ cm}$ and $6\text{ cm}$ respectively. Find the sides $AB$ and $AC$.

Step 1: Identify Tangent Properties

When a circle is inscribed in a triangle, the lengths of tangents drawn from a vertex to the circle are equal. We are given that the circle touches side $BC$ at point $D$. Let's assume it touches $AB$ at $E$ and $AC$ at $F$. Therefore, $BD = BE$, $CD = CF$, and $AE = AF$. We are given $BD = 8\text{ cm}$ and $CD = 6\text{ cm}$. Let $AE = AF = x$.

Step 2: Express Side Lengths

Now we can express the lengths of the sides of the triangle in terms of $x$. Side $AB$ is the sum of $AE$ and $EB$, which is $x + 8$. Side $AC$ is the sum of $AF$ and $FC$, which is $x + 6$. Side $BC$ is the sum of $BD$ and $DC$, which is $8 + 6 = 14\text{ cm}$.

Step 3: Calculate Semi-perimeter

To use Heron's formula, we first need to calculate the semi-perimeter, denoted by $s$. The semi-perimeter is half the sum of the lengths of all three sides of the triangle. Substituting the expressions for $AB$, $AC$, and $BC$, we find $s = x + 14$.

Step 4: Calculate Area using Heron's Formula

Now we apply Heron's formula to find the area of triangle $ABC$. Heron's formula states that the area is the square root of $s(s-a)(s-b)(s-c)$, where $a, b, c$ are the side lengths. Substituting the values, we get the area in terms of $x$.

Step 5: Calculate Area using Inradius Formula

Alternatively, the area of a triangle can also be calculated using the formula $\text{Area} = r \cdot s$, where $r$ is the inradius (radius of the inscribed circle) and $s$ is the semi-perimeter. We are given $r = 4\text{ cm}$ and we found $s = x+14$.

Step 6: Equate Areas and Solve for x

Now we equate the two expressions for the area of triangle $ABC$ and solve for $x$. Squaring both sides eliminates the square root. We can then simplify the equation by dividing by $16(x+14)$ (since $x+14$ cannot be zero for a valid triangle). This leads to a linear equation for $x$, which gives $x=7$.

Step 7: Find Side Lengths AB and AC

Finally, we substitute the value of $x=7$ back into the expressions for the side lengths $AB$ and $AC$. This gives us $AB = 7 + 8 = 15\text{ cm}$ and $AC = 7 + 6 = 13\text{ cm}$.

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