`DeltaABC` is right angled at A and `AD bot BC`. Then `(BD)/(DC)=`

Options

A. `((AB)/(AC))^(2)`
B. `(AB)/(AC)`
C. `((AB)/(AD))^(2)`
D. `(AB)/(AD)`

Answer

To solve the problem, we need to find the ratio \( \frac{BD}{DC} \) in triangle \( \Delta ABC \) where \( A \) is the right angle and \( AD \) is perpendicular to \( BC \).

### Step-by-Step Solution:

1. **Draw the Triangle**:
- Begin by sketching triangle \( ABC \) with \( A \) at the right angle. Label the points as follows: \( A \) (right angle), \( B \), and \( C \).
- Draw a line segment \( BC \) and mark point \( D \) on \( BC \) such that \( AD \) is perpendicular to \( BC \).

**Hint**: Visualizing the problem with a diagram helps in understanding the relationships between the angles and sides.

2. **Identify Similar Triangles**:
- Observe that triangles \( ABD \) and \( ACD \) are formed by the altitude \( AD \).
- The angles in these triangles can be noted: \( \angle ADB = \angle ABC \) (common angle), \( \angle ABD = \angle ACD \) (both are right angles).

**Hint**: Look for common angles to establish similarity between triangles.

3. **Apply AA Similarity Criterion**:
- Since both triangles share angle \( A \) and both have a right angle, by the AA (Angle-Angle) similarity criterion, we can conclude that \( \triangle ABD \sim \triangle ACD \).

**Hint**: Remember that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

4. **Set Up Proportions**:
- From the similarity of the triangles, we can write the proportion:
\[ \frac{AB}{AC} = \frac{BD}{DC} \]

**Hint**: When triangles are similar, the ratios of their corresponding sides are equal.

5. **Cross-Multiply**:
- Rearranging the proportion gives:
\[ BD = \frac{AB}{AC} \cdot DC \]

**Hint**: Cross-multiplication is a useful technique to manipulate proportions.

6. **Express \( \frac{BD}{DC} \)**:
- To find \( \frac{BD}{DC} \), we can rearrange the equation:
\[ \frac{BD}{DC} = \frac{AB}{AC} \]

**Hint**: Isolate the variable of interest to find the desired ratio.

7. **Conclusion**:
- The final result is:
\[ \frac{BD}{DC} = \frac{AB}{AC} \]

**Hint**: Always double-check your final answer against the problem requirements.

### Final Answer:
The ratio \( \frac{BD}{DC} \) is equal to \( \frac{AB}{AC} \).