In $\triangle ABC$, $AD$ is the bisector of $\angle A$, meeting side $BC$ at $D$. If $AB = 12\text{ cm}$, $AC = 18\text{ cm}$, and $BC = 10\text{ cm}$, find the lengths of segments $BD$ and $DC$.

Step 1: Apply the Angle Bisector Theorem

The Angle Bisector Theorem states that if a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle. In $\triangle ABC$, since $AD$ bisects $\angle A$, the ratio of $BD$ to $DC$ is equal to the ratio of $AB$ to $AC$.

Step 2: Substitute given values

We are given the lengths of sides $AB = 12\text{ cm}$ and $AC = 18\text{ cm}$. We substitute these values into the proportion derived from the Angle Bisector Theorem.

Step 3: Simplify the ratio

We simplify the ratio $\frac{12}{18}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 6. This gives us a simplified ratio of $\frac{2}{3}$.

Step 4: Express BD in terms of DC

From the simplified ratio $\frac{BD}{DC} = \frac{2}{3}$, we can express $BD$ in terms of $DC$ by multiplying both sides by $DC$. This relationship will be used in the next step.

Step 5: Use the total length of BC

We know that point $D$ lies on side $BC$, so the sum of the lengths of segments $BD$ and $DC$ must be equal to the total length of $BC$. We are given $BC = 10\text{ cm}$. We substitute $BD = \frac{2}{3} DC$ into the equation $BD + DC = 10$ and solve for $DC$.

Step 6: Solve for DC and BD

To find $DC$, we multiply both sides of the equation $\frac{5}{3} DC = 10$ by $\frac{3}{5}$. Once $DC$ is found, we substitute its value back into the equation $BD = \frac{2}{3} DC$ to find $BD$.

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