In the adjacent figure of $\triangle ABC$, $AD$ is the internal bisector of $\angle A$. If $AB=6$ cm, $BD=3$ cm and $DC=2$ cm, then $AC=$
Show Hint
Whenever an internal angle bisector divides the opposite side into two segments, immediately apply the Angle Bisector Theorem:
\[
\frac{AB}{AC}
=
\frac{BD}{DC}.
\]
Then substitute the known values and solve the resulting proportion. This is one of the fastest and most reliable methods for solving triangle geometry problems involving angle bisectors.
Concept:
The given problem is based on one of the most important results in triangle geometry, namely the Angle Bisector Theorem.
The theorem states that if a line segment drawn from a vertex of a triangle bisects the angle at that vertex and meets the opposite side, then it divides the opposite side in the ratio of the two adjacent sides.
For a triangle $ABC$, if $AD$ is the internal bisector of $\angle A$ and intersects side $BC$ at point $D$, then
\[
\frac{AB}{AC}
=
\frac{BD}{DC}.
\]
This theorem allows us to determine an unknown side length when the other side lengths and divided segments are known.
Step 1: Write down the given information.
From the problem statement, we have:
\[
AB = 6 \text{ cm}
\]
\[
BD = 3 \text{ cm}
\]
\[
DC = 2 \text{ cm}
\]
Also,
\[
AD
\]
is the internal bisector of
\[
\angle A.
\]
We are required to find the length of side
\[
AC.
\]
Step 2: Apply the Angle Bisector Theorem.
Since $AD$ bisects $\angle A$, by the Angle Bisector Theorem,
\[
\frac{AB}{AC}
=
\frac{BD}{DC}.
\]
Substituting the known values,
\[
\frac{6}{AC}
=
\frac{3}{2}.
\]
This equation contains only one unknown quantity, namely $AC$.
Step 3: Solve the proportion.
Cross-multiplying,
\[
6 \times 2
=
3 \times AC.
\]
Therefore,
\[
12
=
3AC.
\]
Dividing both sides by $3$,
\[
AC
=
\frac{12}{3}.
\]
Hence,
\[
AC = 4 \text{ cm}.
\]
Step 4: Verify the result.
Substituting $AC=4$ into the theorem:
\[
\frac{AB}{AC}
=
\frac{6}{4}
=
\frac{3}{2}.
\]
Also,
\[
\frac{BD}{DC}
=
\frac{3}{2}.
\]
Both ratios are equal.
Therefore, the calculated value satisfies the Angle Bisector Theorem and is correct.
Final Answer:
\[
\boxed{AC = 4 \text{ cm}}
\]
Hence, the correct option is
\[
\boxed{\text{(A) 4 cm}}.
\]
