find the height of an equlateral triangle having each side 12 cm.

Answer

To find the height of an equilateral triangle with each side measuring 12 cm, we can follow these steps:

### Step 1: Identify the triangle and its properties
Let triangle ABC be an equilateral triangle where each side is 12 cm. We will drop a perpendicular from vertex A to the midpoint D of side BC. This perpendicular AD will represent the height of the triangle.

### Step 2: Determine the length of BD
Since D is the midpoint of BC, we can find the length of BD. Since BC = 12 cm, we have:
\[ BD = \frac{BC}{2} = \frac{12}{2} = 6 \text{ cm} \]

### Step 3: Apply the Pythagorean theorem
In triangle ABD, we can apply the Pythagorean theorem. According to the theorem:
\[ AB^2 = AD^2 + BD^2 \]
Here, AB is the side of the triangle (12 cm), AD is the height we want to find, and BD is 6 cm.

### Step 4: Substitute the known values into the equation
Substituting the known values into the equation gives us:
\[ 12^2 = AD^2 + 6^2 \]
This simplifies to:
\[ 144 = AD^2 + 36 \]

### Step 5: Solve for AD^2
Now, we can isolate AD^2:
\[ AD^2 = 144 - 36 \]
\[ AD^2 = 108 \]

### Step 6: Calculate AD
To find AD, we take the square root of both sides:
\[ AD = \sqrt{108} \]
We can simplify this further:
\[ AD = \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3} \text{ cm} \]

### Conclusion
Thus, the height of the equilateral triangle is:
\[ \text{Height} = 6\sqrt{3} \text{ cm} \]

---