D is a point on the side BC of `triangle ABC` such that `angle ADC = angle BAC`. If CA = 12 cm and BC = 16 cm, then the length of CD is :

Options

A. 12 cm
B. 6 cm
C. 8 cm
D. 9 cm

Answer

To solve the problem, we need to find the length of segment CD in triangle ABC, given that angle ADC is equal to angle BAC, and we know the lengths of CA and BC.

### Step-by-step Solution:

1. **Identify the Given Information**:
- CA = 12 cm
- BC = 16 cm
- Angle ADC = Angle BAC

2. **Use the Angle-Angle (AA) Similarity Criterion**:
- Since angle ADC = angle BAC, triangles ADC and ABC are similar by the AA criterion.

3. **Set Up the Proportions Based on Similarity**:
- From the similarity of triangles, we can set up the following ratios:
\[ \frac{AB}{AD} = \frac{BC}{AC} = \frac{AC}{CD} \]

4. **Substitute the Known Values**:
- We know BC = 16 cm and AC = 12 cm. We need to find CD.
- Using the ratio \(\frac{BC}{AC} = \frac{16}{12}\):
\[ \frac{16}{12} = \frac{12}{CD} \]

5. **Cross-Multiply to Solve for CD**:
- Cross-multiplying gives:
\[ 16 \cdot CD = 12 \cdot 12 \]
\[ 16 \cdot CD = 144 \]

6. **Isolate CD**:
- Divide both sides by 16:
\[ CD = \frac{144}{16} = 9 \text{ cm} \]

### Final Answer:
The length of CD is **9 cm**.