The height of an equilateral triangle measures 9 cm. Find its area, correct to 2 places of decimal. (Take `sqrt(3) = 1.732`.)

Explanation

To find the area of an equilateral triangle given its height, we can follow these steps:

### Step 1: Understand the properties of an equilateral triangle
In an equilateral triangle, all sides are equal, and the altitude (height) from any vertex to the opposite side bisects that side. This means that if we drop a perpendicular from one vertex to the base, it divides the base into two equal segments.

### Step 2: Set up the triangle
Let the equilateral triangle be ABC, where:
- A is the vertex from which the height is dropped.
- D is the point where the height meets the base BC.
- The height AD = 9 cm.

Since AD is the height, it divides the base BC into two equal segments. Let the length of each segment (BD and DC) be \( \frac{A}{2} \), where A is the length of the side of the triangle.

### Step 3: Apply the Pythagorean theorem
In triangle ABD, we can apply the Pythagorean theorem:
\[ AB^2 = AD^2 + BD^2 \]
Substituting the known values:
\[ A^2 = 9^2 + \left(\frac{A}{2}\right)^2 \]
This simplifies to:
\[ A^2 = 81 + \frac{A^2}{4} \]

### Step 4: Rearrange the equation
To eliminate the fraction, multiply the entire equation by 4:
\[ 4A^2 = 324 + A^2 \]
Now, rearranging gives:
\[ 4A^2 - A^2 = 324 \]
\[ 3A^2 = 324 \]

### Step 5: Solve for A^2
Dividing both sides by 3:
\[ A^2 = \frac{324}{3} = 108 \]

### Step 6: Calculate the area of the triangle
The area \( A \) of an equilateral triangle can be calculated using the formula:
\[ \text{Area} = \frac{\sqrt{3}}{4} A^2 \]
Substituting \( A^2 = 108 \):
\[ \text{Area} = \frac{\sqrt{3}}{4} \times 108 \]
Using \( \sqrt{3} \approx 1.732 \):
\[ \text{Area} = \frac{1.732}{4} \times 108 = 27 \times 1.732 \]
Calculating this gives:
\[ \text{Area} \approx 46.764 \text{ cm}^2 \]

### Step 7: Round to two decimal places
Rounding to two decimal places, the area of the triangle is:
\[ \text{Area} \approx 46.76 \text{ cm}^2 \]

### Final Answer
The area of the equilateral triangle is **46.76 cm²**.