Find the height of an equilateral Delta whose side is ` 2sqrt(3)` cm .

Answer

To find the height of an equilateral triangle (delta) with a side length of \( 2\sqrt{3} \) cm, we can follow these steps:

### Step 1: Understand the Properties of the Equilateral Triangle
An equilateral triangle has all three sides equal and all three angles equal to \( 60^\circ \). The height can be found by dropping a perpendicular from one vertex to the opposite side, which bisects the base.

### Step 2: Label the Triangle
Let the equilateral triangle be \( ABC \) with each side measuring \( 2\sqrt{3} \) cm. We will drop a perpendicular from vertex \( A \) to the midpoint \( D \) of side \( BC \).

### Step 3: Find the Length of \( BD \)
Since \( D \) is the midpoint of \( BC \), we can find \( BD \) as follows:
\[ BD = \frac{BC}{2} = \frac{2\sqrt{3}}{2} = \sqrt{3} \text{ cm} \]

### Step 4: Use the Pythagorean Theorem
In triangle \( ABD \), we can apply the Pythagorean theorem:
\[ AB^2 = AD^2 + BD^2 \]
Where:
- \( AB = 2\sqrt{3} \)
- \( BD = \sqrt{3} \)
- \( AD \) is the height we want to find.

Substituting the known values:
\[ (2\sqrt{3})^2 = AD^2 + (\sqrt{3})^2 \]
\[ 12 = AD^2 + 3 \]

### Step 5: Solve for \( AD^2 \)
Rearranging the equation gives:
\[ AD^2 = 12 - 3 = 9 \]

### Step 6: Find \( AD \)
Taking the square root of both sides:
\[ AD = \sqrt{9} = 3 \text{ cm} \]

### Conclusion
The height of the equilateral triangle is \( 3 \) cm.

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