D is a point on the side BC of a `angleABC` such that `angleADC=angleBAC`. If CA = 10 cm and BC = 16 cm then the length of CD is:

Options

A. `6.25` cm
B. `6.5` cm
C. 7 cm
D. 6 cm

Answer

To solve the problem, we will use the properties of similar triangles. Here are the steps to find the length of CD:

### Step 1: Identify the triangles
We have triangle ABC and point D on side BC such that angle ADC = angle BAC. This means triangles ADC and ABC are similar by the AA (Angle-Angle) similarity criterion.

### Step 2: Set up the similarity ratio
From the similarity of triangles, we can write the following ratio:
\[ \frac{AB}{AC} = \frac{AD}{AB} = \frac{BC}{CD} \]
We will focus on the sides involving CD and the known lengths.

### Step 3: Assign known values
From the problem statement:
- AC = 10 cm
- BC = 16 cm

### Step 4: Set up the equation using the similarity ratio
Using the similarity ratio:
\[ \frac{BC}{AC} = \frac{CD}{AC} \]
Substituting the known values:
\[ \frac{16}{10} = \frac{CD}{10} \]

### Step 5: Solve for CD
Cross-multiplying gives:
\[ 16 \cdot CD = 10 \cdot 10 \]
\[ 16 \cdot CD = 100 \]
Now, divide both sides by 16:
\[ CD = \frac{100}{16} = 6.25 \text{ cm} \]

### Final Answer
Thus, the length of CD is **6.25 cm**.