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 then the sum of sides
D. twice the sum of sides.

Answer

To solve the question regarding the sum of the three altitudes of a triangle, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Triangle and Altitudes**:
- Let's denote the triangle as \( ABC \).
- The altitudes from vertices \( A, B, \) and \( C \) to the opposite sides are denoted as \( h_a, h_b, \) and \( h_c \) respectively.

2. **Drawing the Triangle**:
- Draw triangle \( ABC \) and label the vertices.
- Draw the altitudes \( h_a \) from vertex \( A \) to side \( BC \), \( h_b \) from vertex \( B \) to side \( AC \), and \( h_c \) from vertex \( C \) to side \( AB \).

3. **Applying the Pythagorean Theorem**:
- For each altitude, we can apply the Pythagorean theorem to relate the sides of the triangle to the altitudes.
- For triangle \( ABL \) (where \( L \) is the foot of the altitude from \( A \)):
\[ AB^2 = AL^2 + BL^2 \quad \text{(where \( AL = h_a \))} \]
This implies \( AB^2 > h_a^2 \) which leads to \( AB > h_a \).

4. **Repeating for Other Altitudes**:
- Similarly, for triangle \( CAN \) (where \( N \) is the foot of the altitude from \( C \)):
\[ AC^2 = CN^2 + AN^2 \quad \text{(where \( CN = h_c \))} \]
This implies \( AC^2 > h_c^2 \) which leads to \( AC > h_c \).

- For triangle \( BMC \) (where \( M \) is the foot of the altitude from \( B \)):
\[ BC^2 = BM^2 + CM^2 \quad \text{(where \( BM = h_b \))} \]
This implies \( BC^2 > h_b^2 \) which leads to \( BC > h_b \).

5. **Summing the Inequalities**:
- Now, we have three inequalities:
\[ AB > h_a, \quad AC > h_c, \quad BC > h_b \]
- Adding these inequalities gives:
\[ AB + AC + BC > h_a + h_b + h_c \]

6. **Conclusion**:
- Therefore, the sum of the altitudes \( h_a + h_b + h_c \) is less than the sum of the sides \( AB + AC + BC \).
- This means that 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 of the triangle.