Find the length of the altitude of an equilateral triangle of side `9sqrt3 cm`.

Options

A. 21 cm
B. 15.5 cm
C. 14 cm
D. 13.5 cm

Answer

To find the length of the altitude of an equilateral triangle with a side length of \(9\sqrt{3}\) cm, we can follow these steps:

### Step 1: Understand the properties of an equilateral triangle
An equilateral triangle has all three sides equal, and the altitude divides the triangle into two 30-60-90 right triangles.

### Step 2: Identify the side length
Given the side length of the equilateral triangle is \(9\sqrt{3}\) cm.

### Step 3: Calculate the length of half the base
The altitude bisects the base of the triangle. Therefore, the length of half the base (which we will denote as \(BO\)) is:
\[ BO = \frac{9\sqrt{3}}{2} \text{ cm} \]

### Step 4: Apply the Pythagorean theorem
In triangle \(AOB\), we can apply the Pythagorean theorem:
\[ AB^2 = AO^2 + BO^2 \]
Where:
- \(AB\) is the hypotenuse (which is the side of the triangle, \(9\sqrt{3}\) cm),
- \(AO\) is the altitude we want to find,
- \(BO\) is half the base, which we calculated as \(\frac{9\sqrt{3}}{2}\).

### Step 5: Substitute the values into the equation
Substituting the known values into the Pythagorean theorem:
\[ (9\sqrt{3})^2 = AO^2 + \left(\frac{9\sqrt{3}}{2}\right)^2 \]

### Step 6: Calculate the squares
Calculating the squares:
\[ (9\sqrt{3})^2 = 81 \times 3 = 243 \]
\[ \left(\frac{9\sqrt{3}}{2}\right)^2 = \frac{(9\sqrt{3})^2}{4} = \frac{243}{4} \]

### Step 7: Set up the equation
Now we can set up the equation:
\[ 243 = AO^2 + \frac{243}{4} \]

### Step 8: Isolate \(AO^2\)
To isolate \(AO^2\), we subtract \(\frac{243}{4}\) from both sides:
\[ AO^2 = 243 - \frac{243}{4} \]
To perform this subtraction, convert \(243\) into a fraction with a denominator of \(4\):
\[ 243 = \frac{972}{4} \]
So,
\[ AO^2 = \frac{972}{4} - \frac{243}{4} = \frac{729}{4} \]

### Step 9: Take the square root
Now, take the square root of both sides to find \(AO\):
\[ AO = \sqrt{\frac{729}{4}} = \frac{\sqrt{729}}{\sqrt{4}} = \frac{27}{2} = 13.5 \text{ cm} \]

### Final Answer
The length of the altitude of the equilateral triangle is \(13.5\) cm.

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