The sum of three altitudes of a triangle is

Options

A. equal to the sum of three sides
B. less than the sum of sides
C. greater than the sum of sides
D. twice the sum of sides

Answer

To solve the problem of the sum of the three altitudes of a triangle, we will analyze the relationship between the altitudes and the sides of the triangle.

### Step-by-Step Solution:

1. **Understanding Altitudes**:
The altitude of a triangle is the perpendicular distance from a vertex to the line containing the opposite side. For a triangle with vertices A, B, and C, the altitudes can be denoted as h_a (from A to BC), h_b (from B to AC), and h_c (from C to AB).

2. **Triangle Properties**:
In any triangle, the lengths of the sides are always greater than the lengths of the corresponding altitudes. This is because the altitude represents the shortest distance from a vertex to the opposite side.

3. **Comparing Altitudes with Sides**:
Let's denote the sides of the triangle as a (BC), b (AC), and c (AB). According to the properties of triangles:
- h_a < b
- h_a < c
- h_b < a
- h_b < c
- h_c < a
- h_c < b

4. **Summing the Altitudes**:
When we sum the three altitudes, we have:
\[ h_a + h_b + h_c \]
Since each altitude is less than the corresponding sides, we can conclude:
\[ h_a + h_b + h_c < a + b + c \]

5. **Conclusion**:
Therefore, the sum of the three altitudes of a triangle is less than the sum of the three sides of the triangle.

### Final Answer:
The sum of the three altitudes of a triangle is **less than the sum of the three sides**.

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