The altitude of an equilateral triangle having the length of its side 10 cm is `Ksqrt(3)` cm . Find K .

Answer

To solve the problem of finding the value of \( K \) in the altitude of an equilateral triangle with a side length of 10 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the triangle**: We have an equilateral triangle \( ABC \) where each side \( AB = BC = CA = 10 \) cm.

2. **Draw the altitude**: Draw the altitude \( AD \) from vertex \( A \) to the midpoint \( D \) of side \( BC \). Since \( D \) is the midpoint, we have \( BD = DC = 5 \) cm.

3. **Apply the Pythagorean theorem**: In the right triangle \( ABD \), we can apply the Pythagorean theorem:
\[ AB^2 = AD^2 + BD^2 \]
Here, \( AB = 10 \) cm and \( BD = 5 \) cm.

4. **Substitute the known values**:
\[ 10^2 = AD^2 + 5^2 \]
Simplifying this gives:
\[ 100 = AD^2 + 25 \]

5. **Solve for \( AD^2 \)**:
\[ AD^2 = 100 - 25 = 75 \]

6. **Find \( AD \)**:
\[ AD = \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3} \text{ cm} \]

7. **Identify \( K \)**: The altitude \( AD \) is expressed as \( K\sqrt{3} \) cm. From our calculation, we see that:
\[ AD = 5\sqrt{3} \implies K = 5 \]

### Final Answer:
Thus, the value of \( K \) is \( 5 \).

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