Let `AX_|_BC `of an equilateral triangle ABC. Then the sum of the perpendicular distances of the sides of `DeltaABC `from any point inside the triangle is :

Options

A. Equal to BC
B. Equal to AX
C. Less than AX
D. Greater than AX

Answer

To solve the problem, we need to determine the sum of the perpendicular distances from any point inside an equilateral triangle \( ABC \) to its sides.

### Step-by-Step Solution:

1. **Understanding the Triangle**:
We have an equilateral triangle \( ABC \). In an equilateral triangle, all sides are equal, and all angles are \( 60^\circ \).

2. **Identifying the Point Inside the Triangle**:
Let \( P \) be any point inside the triangle \( ABC \). We need to find the perpendicular distances from point \( P \) to each of the sides \( AB \), \( BC \), and \( CA \).

3. **Denote the Distances**:
Let the perpendicular distances from point \( P \) to the sides \( AB \), \( BC \), and \( CA \) be denoted as \( d_1 \), \( d_2 \), and \( d_3 \) respectively.

4. **Using Area to Relate Distances**:
The area \( A \) of triangle \( ABC \) can be expressed in two ways:
- Using the base and height:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
- Using the perpendicular distances from point \( P \):
\[ A = \frac{1}{2} \times AB \times d_1 + \frac{1}{2} \times BC \times d_2 + \frac{1}{2} \times CA \times d_3 \]

5. **Equilateral Triangle Properties**:
Since \( ABC \) is equilateral, \( AB = BC = CA \). Let's denote the length of each side as \( s \). Thus, we can write:
\[ A = \frac{s \cdot h}{2} \]
where \( h \) is the height of the triangle.

6. **Relating Areas**:
The area can also be expressed as:
\[ A = \frac{s \cdot d_1}{2} + \frac{s \cdot d_2}{2} + \frac{s \cdot d_3}{2} \]
Simplifying this gives:
\[ A = \frac{s}{2} (d_1 + d_2 + d_3) \]

7. **Equating the Two Area Expressions**:
Since both expressions represent the same area \( A \), we can set them equal:
\[ \frac{s \cdot h}{2} = \frac{s}{2} (d_1 + d_2 + d_3) \]

8. **Canceling Common Terms**:
By canceling \( \frac{s}{2} \) (assuming \( s \neq 0 \)):
\[ h = d_1 + d_2 + d_3 \]

9. **Conclusion**:
The sum of the perpendicular distances from any point inside the triangle \( ABC \) to its sides is equal to the height \( h \) of the triangle.

### Final Answer:
The sum of the perpendicular distances from any point inside the triangle \( ABC \) to its sides is equal to the height of the triangle.