A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that BD = 8 cm and DC = 6 cm. Find the sides AB and AC.

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Step-by-Step Solution

Step 1: Identify Tangent Properties

According to the property of tangents drawn from an external point to a circle, the lengths of the tangents from the external point to the circle are equal. Here, the vertices A, B, and C are external points. So, $BD = BF$, $CD = CE$, and $AE = AF$. We are given $BD = 8 \text{ cm}$ and $CD = 6 \text{ cm}$. Let $AE = AF = x.

Step 2: Express Sides in terms of x

Now we can express the lengths of the sides of the triangle ABC in terms of $x$. Side $AB$ is the sum of $AF$ and $FB$, which is $x + 8$. Side $AC$ is the sum of $AE$ and $EC$, 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 (s)

The semi-perimeter $s$ of the triangle is half the sum of its sides. Substituting the expressions for $AB$, $BC$, and $AC$, we get $s = \frac{(x+8) + 14 + (x+6)}{2}$, which simplifies to $s = x + 14.

Step 4: Calculate Area using Heron's Formula

We use Heron's formula to calculate the area of triangle ABC. Substituting the values of $s$, $AB$, $BC$, and $AC$ into the formula $\sqrt{s(s-AB)(s-BC)(s-AC)}$, we simplify the expression to $\sqrt{48x(x+14)}.

Step 5: Calculate Area using Radius and Semi-perimeter

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

Step 6: Equate Areas and Solve for x

By equating the two expressions for the area of triangle ABC, we get $\sqrt{48x(x+14)} = 4(x+14)$. Squaring both sides and simplifying, we arrive at $3x = x+14$, which gives $2x = 14$, so $x = 7$. We can divide by $(x+14)$ because $x$ represents a length, so $x > 0$, making $x+14 eq 0.

Step 7: Calculate Sides AB and AC

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