The length of an altitude in an equilateral triangle of side 'o' cm is ……………. .

Answer

To find the length of an altitude in an equilateral triangle with side length 'O' cm, we can follow these steps:

### Step-by-Step Solution:

1. **Draw the Equilateral Triangle**:
Let's denote the equilateral triangle as \( ABC \) where \( AB = BC = CA = O \) cm.

2. **Identify the Midpoint**:
Let \( D \) be the midpoint of side \( BC \). Since \( D \) is the midpoint, we have \( BD = DC = \frac{O}{2} \) cm.

3. **Draw the Altitude**:
Draw the altitude \( AD \) from vertex \( A \) to the midpoint \( D \) of side \( BC \). The altitude \( AD \) is perpendicular to \( BC \).

4. **Apply the Pythagorean Theorem**:
In right triangle \( ADB \):
- The hypotenuse \( AB = O \) cm.
- One leg \( BD = \frac{O}{2} \) cm.
- The other leg \( AD \) is the altitude we want to find, which we will denote as \( x \).

According to the Pythagorean theorem:
\[ AB^2 = AD^2 + BD^2 \]
Substituting the known values:
\[ O^2 = x^2 + \left(\frac{O}{2}\right)^2 \]

5. **Simplify the Equation**:
Expanding the equation:
\[ O^2 = x^2 + \frac{O^2}{4} \]

6. **Rearranging the Equation**:
To isolate \( x^2 \), subtract \( \frac{O^2}{4} \) from both sides:
\[ O^2 - \frac{O^2}{4} = x^2 \]
This simplifies to:
\[ \frac{4O^2}{4} - \frac{O^2}{4} = x^2 \]
\[ \frac{3O^2}{4} = x^2 \]

7. **Taking the Square Root**:
Now, take the square root of both sides to find \( x \):
\[ x = \sqrt{\frac{3O^2}{4}} = \frac{\sqrt{3}O}{2} \]

8. **Conclusion**:
Thus, the length of the altitude in an equilateral triangle with side length \( O \) cm is:
\[ \frac{\sqrt{3}O}{2} \text{ cm} \]