ABC is a right angled triangle At B such that AB = a-b, BC = a and CA = a + 6.D is a point on BC such that BD = AB. The ratio of BD : BC for any value of a and b is given by

Options

A. `3 : 2`
B. `4 : 3`
C. `5 : 4`
D. `3 : 1`

Answer

To solve the problem step by step, we will analyze the given triangle ABC and the relationships between its sides.

### Step 1: Identify the lengths of the sides
Given:
- \( AB = a - b \)
- \( BC = a \)
- \( CA = a + 6 \)

### Step 2: Determine the length of segment BD
Since point D is on BC such that \( BD = AB \), we have:
- \( BD = a - b \)

### Step 3: Calculate the length of segment DC
To find \( DC \), we can use the relationship:
\[ BC = BD + DC \]
Substituting the known values:
\[ a = (a - b) + DC \]
Rearranging gives:
\[ DC = a - (a - b) = b \]

### Step 4: Find the ratio of BD to BC
Now we need to find the ratio \( \frac{BD}{BC} \):
\[ \frac{BD}{BC} = \frac{a - b}{a} \]

### Step 5: Simplify the ratio
To simplify \( \frac{a - b}{a} \):
\[ \frac{BD}{BC} = 1 - \frac{b}{a} \]

### Step 6: Conclusion
The ratio \( \frac{BD}{BC} \) can be expressed as \( 1 - \frac{b}{a} \). However, we are interested in the specific ratio of \( BD : DC \).

From our earlier calculations:
- \( BD = a - b \)
- \( DC = b \)

Thus, the ratio \( BD : DC \) is:
\[ \frac{BD}{DC} = \frac{a - b}{b} \]

### Step 7: Final Ratio
To express this in a simpler form, we can rewrite it as:
\[ \frac{a - b}{b} = \frac{a}{b} - 1 \]