In an equilateral `triangle ABC`, if a, b and c denote the lengths of perpendiculars from A, B and C respectively on the opposite sides, then :

Options

A. `agtbgtc`
B. `agtbltc`
C. a=b=c
D. `a=cneb`

Answer

To solve the problem, we need to analyze the properties of an equilateral triangle and the lengths of the perpendiculars dropped from each vertex to the opposite side.

### Step-by-Step Solution:

1. **Understanding the Equilateral Triangle**:
In an equilateral triangle, all sides are equal, and all angles are equal (each measuring 60 degrees). Let's denote the side length of triangle ABC as \( s \).

2. **Identifying the Perpendiculars**:
Let \( A, B, C \) denote the lengths of the perpendiculars from vertices \( A, B, C \) to the opposite sides \( BC, CA, AB \) respectively. We will denote these perpendiculars as \( AD, BE, CF \) where \( D, E, F \) are the feet of the perpendiculars from \( A, B, C \) respectively.

3. **Calculating the Length of the Perpendiculars**:
The height (or altitude) of an equilateral triangle can be calculated using the formula:
\[ h = \frac{\sqrt{3}}{2} s \]
Since the triangle is equilateral, the lengths of the perpendiculars from each vertex to the opposite side will be equal.

4. **Setting Up the Equalities**:
Since \( A, B, C \) are the lengths of the perpendiculars from vertices \( A, B, C \) respectively, we can conclude that:
\[ A = h, \quad B = h, \quad C = h \]
Therefore, we can write:
\[ A = B = C \]

5. **Conclusion**:
The lengths of the perpendiculars from each vertex to the opposite side in an equilateral triangle are equal. Thus, we conclude:
\[ A = B = C \]

### Final Answer:
The lengths of the perpendiculars from the vertices of an equilateral triangle to the opposite sides are equal, i.e., \( A = B = C \).

---