In `Delta ABC, AD and BE` are perpendiculars from A and B to the sides BC and AC, then

Options

A. CA. CE = CB. CD
B. CB. CA = CE. CD
C. `(AD)/(BE) = (AB)/(DE)`
D. None of these

Answer

To solve the problem, we need to analyze the triangle ABC with perpendiculars AD and BE drawn from points A and B to the sides BC and AC, respectively. We will show that triangles ADC and BEC are similar and derive the required relationship.

### Step-by-Step Solution:

1. **Draw Triangle ABC**:
- Start by sketching triangle ABC with vertices A, B, and C. Label the points accordingly.

2. **Draw Perpendiculars AD and BE**:
- From point A, draw a perpendicular line to side BC, and label the foot of the perpendicular as point D.
- From point B, draw a perpendicular line to side AC, and label the foot of the perpendicular as point E.

3. **Identify Angles**:
- In triangle ADC, we have:
- Angle A = θ
- Angle D = 90° (since AD is perpendicular to BC)
- Therefore, Angle C = 90° - θ (by the angle sum property of triangles).
- In triangle BEC, we have:
- Angle B = θ
- Angle E = 90° (since BE is perpendicular to AC)
- Therefore, Angle C = 90° - θ (by the angle sum property of triangles).

4. **Establish Similarity**:
- Since both triangles ADC and BEC have two angles equal (Angle C is common, and Angle A = Angle B = θ), we can conclude that triangle ADC is similar to triangle BEC by the AA (Angle-Angle) criterion for similarity.

5. **Set Up Proportions**:
- From the similarity of triangles, we can write the following proportion:
\[ \frac{AD}{BE} = \frac{CD}{CB} \]
- Rearranging gives us:
\[ AD \cdot CB = BE \cdot CD \]

6. **Final Relationship**:
- The relationship we derived shows that the product of the lengths of the segments created by the perpendiculars is equal, which is the required relationship.

### Conclusion:
The relationship derived from the similarity of triangles ADC and BEC is:
\[ AD \cdot CB = BE \cdot CD \]