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

Options

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

Answer

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

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

### Step 2: Substitute the given height into the formula to find the side length.
Given that the height \( h = 15 \) cm, we can rearrange the formula to solve for \( a \):
\[ 15 = \frac{\sqrt{3}}{2} a \]
To isolate \( a \), multiply both sides by \( \frac{2}{\sqrt{3}} \):
\[ a = \frac{15 \times 2}{\sqrt{3}} = \frac{30}{\sqrt{3}} \]

### Step 3: Rationalize the side length.
To rationalize \( a \):
\[ a = \frac{30}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{30\sqrt{3}}{3} = 10\sqrt{3} \text{ cm} \]

### Step 4: Use the area formula for the triangle.
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
For an equilateral triangle, the base is equal to the side length \( a \):
\[ A = \frac{1}{2} \times a \times h \]

### Step 5: Substitute the values of \( a \) and \( h \) into the area formula.
Substituting \( a = 10\sqrt{3} \) cm and \( h = 15 \) cm:
\[ A = \frac{1}{2} \times (10\sqrt{3}) \times 15 \]
\[ A = \frac{1}{2} \times 150\sqrt{3} \]
\[ A = 75\sqrt{3} \text{ cm}^2 \]

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

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