Each side of an equilateral triangle is equal to the radius of a circle whose area is 154 cm . The area of the triangle is

Options

A. `(7sqrt3)/(4)cm^(2)`
B. `(49sqrt3)/(4)cm^(2)`
C. `35cm^(2)`
D. `49cm^(2)`

Answer

To solve the problem step by step, we will first find the radius of the circle and then use it to find the area of the equilateral triangle.

### Step 1: Find the radius of the circle
The area of the circle is given as 154 cm². The formula for the area of a circle is:
\[ \text{Area} = \pi r^2 \]
Substituting the known area into the formula:
\[ 154 = \pi r^2 \]
Using \(\pi \approx \frac{22}{7}\), we can rewrite the equation:
\[ 154 = \frac{22}{7} r^2 \]

### Step 2: Solve for \(r^2\)
To isolate \(r^2\), we multiply both sides by \(\frac{7}{22}\):
\[ r^2 = 154 \times \frac{7}{22} \]
Calculating the right side:
\[ r^2 = \frac{154 \times 7}{22} \]
First, simplify \(\frac{154}{22}\):
\[ \frac{154}{22} = 7 \]
So,
\[ r^2 = 7 \times 7 = 49 \]

### Step 3: Find \(r\)
Now, take the square root of both sides to find \(r\):
\[ r = \sqrt{49} = 7 \text{ cm} \]

### Step 4: Find the side of the triangle
According to the problem, each side of the equilateral triangle is equal to the radius of the circle:
\[ \text{Side of triangle} (s) = r = 7 \text{ cm} \]

### Step 5: Calculate the area of the equilateral triangle
The formula for the area of an equilateral triangle is:
\[ \text{Area} = \frac{\sqrt{3}}{4} s^2 \]
Substituting \(s = 7\):
\[ \text{Area} = \frac{\sqrt{3}}{4} (7^2) \]
Calculating \(7^2\):
\[ 7^2 = 49 \]
Now substituting back:
\[ \text{Area} = \frac{\sqrt{3}}{4} \times 49 = \frac{49\sqrt{3}}{4} \text{ cm}^2 \]

### Final Answer
The area of the triangle is:
\[ \frac{49\sqrt{3}}{4} \text{ cm}^2 \]

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