ABC is a right angle triangle at A and AD is perpendicular to the hypotenuse. Then `(BD)/(CD)` is equal to :

Options

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

Answer

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

### Step-by-Step Solution:

1. **Identify the Triangle and Points**:
- We have a right triangle \( ABC \) with \( \angle A = 90^\circ \).
- Point \( D \) is the foot of the perpendicular from \( A \) to the hypotenuse \( BC \).

2. **Use the Property of Right Triangles**:
- In a right triangle, the segments created by the altitude from the right angle to the hypotenuse have a special relationship. Specifically, we can use the following relationships:
\[ \frac{BD}{CD} = \frac{AB^2}{AC^2} \]

3. **Set Up the Ratios**:
- Let \( AB = c \), \( AC = b \), and \( BC = a \) (the hypotenuse).
- According to the property mentioned, we can express \( BD \) and \( CD \) in terms of the sides of the triangle:
\[ BD = \frac{AB^2}{BC} = \frac{c^2}{a} \]
\[ CD = \frac{AC^2}{BC} = \frac{b^2}{a} \]

4. **Calculate the Ratio**:
- Now, substituting these values into the ratio \( \frac{BD}{CD} \):
\[ \frac{BD}{CD} = \frac{\frac{c^2}{a}}{\frac{b^2}{a}} = \frac{c^2}{b^2} \]

5. **Final Result**:
- Therefore, the ratio \( \frac{BD}{CD} \) simplifies to:
\[ \frac{BD}{CD} = \left(\frac{AB}{AC}\right)^2 \]
- This means that \( \frac{BD}{CD} \) is equal to the square of the ratio of the lengths of the sides \( AB \) and \( AC \).

### Conclusion:
Thus, the answer to the question \( \frac{BD}{CD} \) is \( \frac{AB^2}{AC^2} \).

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