The height of an equilateral triangle is 15 cm. The area of the triangle is

Options

A. `50sqrt(3)` sq. cm.
B. `70sqrt(3)` sq. cm.
C. `75 sqrt(3)` sq. cm.
D. `150sqrt(3)` sq. cm.

Answer

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

### Step-by-Step Solution:

1. **Understand the relationship between the height and the side of an equilateral triangle**:
The height (h) of an equilateral triangle can be expressed in terms of its side length (a) using the formula:
\[ h = \frac{a \sqrt{3}}{2} \]

2. **Substitute the given height into the formula**:
We know the height \( h = 15 \, \text{cm} \). Therefore, we can set up the equation:
\[ 15 = \frac{a \sqrt{3}}{2} \]

3. **Solve for the side length (a)**:
To isolate \( a \), multiply both sides by 2:
\[ 30 = a \sqrt{3} \]
Now, divide both sides by \( \sqrt{3} \):
\[ a = \frac{30}{\sqrt{3}} \]

4. **Rationalize the denominator**:
To simplify \( a \), multiply the numerator and the denominator by \( \sqrt{3} \):
\[ a = \frac{30 \sqrt{3}}{3} = 10 \sqrt{3} \, \text{cm} \]

5. **Calculate the area of the equilateral triangle**:
The area \( A \) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Substitute \( a = 10 \sqrt{3} \):
\[ A = \frac{\sqrt{3}}{4} (10 \sqrt{3})^2 \]

6. **Simplify the expression**:
Calculate \( (10 \sqrt{3})^2 \):
\[ (10 \sqrt{3})^2 = 100 \times 3 = 300 \]
Now substitute back into the area formula:
\[ A = \frac{\sqrt{3}}{4} \times 300 = \frac{300 \sqrt{3}}{4} = 75 \sqrt{3} \, \text{cm}^2 \]

### Final Answer:
The area of the equilateral triangle is \( 75 \sqrt{3} \, \text{cm}^2 \).