In $\triangle ABC$, $AD$ is the bisector of $\angle A$, meeting side $BC$ at $D$. If $AB = 10\text{ cm}$, $AC = 14\text{ cm}$, and $BC = 6\text{ cm}$, 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 $\triangle ABC$, $AD$ bisects $\angle A$, so it divides side $BC$ into segments $BD$ and $DC$ such that their ratio is equal to the ratio of the other two sides, $AB$ and $AC$.

Step 2: Substitute given values

We are given the lengths of sides $AB = 10\text{ cm}$ and $AC = 14\text{ cm}$. Substitute these values into the proportion derived from the Angle Bisector Theorem.

Step 3: Simplify the ratio and express BD in terms of DC

Simplify the ratio $\frac{10}{14}$ to $\frac{5}{7}$. Then, rearrange the equation to express $BD$ in terms of $DC$. This will be useful for solving the system of equations.

Step 4: Use the total length of BC

We know that point $D$ lies on the 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 $BC = 6\text{ cm}$.

Step 5: Solve for DC

Substitute the expression for $BD$ from step 3 into the equation from step 4. Combine the terms involving $DC$ and solve for $DC$.

Step 6: Solve for BD

Now that we have the value of $DC$, we can use the equation $BC = BD + DC$ to find the length of $BD$.

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