A piece of wire is in the shape of an equilateral triangle, each of whose sides is 4.4 cm. If it is re-bent to form a circular ring, the radius of the ring so formed is (taking `pi= (22)/(7)`)

Options

A. 5.1
B. 3.2
C. 2.1
D. 1.5

Answer

To find the radius of the circular ring formed by re-bending a piece of wire in the shape of an equilateral triangle, we can follow these steps:

### Step 1: Calculate the perimeter of the equilateral triangle.
The perimeter \( P \) of an equilateral triangle is given by the formula:
\[ P = 3 \times \text{side length} \]
Given that each side of the triangle is \( 4.4 \) cm, we can calculate the perimeter:
\[ P = 3 \times 4.4 = 13.2 \text{ cm} \]

### Step 2: Understand that the perimeter of the triangle is equal to the circumference of the circular ring.
When the wire is re-bent to form a circular ring, the length of the wire remains the same. Therefore, the circumference \( C \) of the circular ring is equal to the perimeter of the triangle:
\[ C = P = 13.2 \text{ cm} \]

### Step 3: Use the circumference formula to find the radius.
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
Where \( r \) is the radius. We can rearrange this formula to solve for \( r \):
\[ r = \frac{C}{2\pi} \]

### Step 4: Substitute the values into the formula.
We know \( C = 13.2 \) cm and \( \pi = \frac{22}{7} \). Now we can substitute these values into the formula for \( r \):
\[ r = \frac{13.2}{2 \times \frac{22}{7}} = \frac{13.2 \times 7}{2 \times 22} \]

### Step 5: Simplify the expression.
Calculating the numerator:
\[ 13.2 \times 7 = 92.4 \]
Calculating the denominator:
\[ 2 \times 22 = 44 \]
Now substituting back:
\[ r = \frac{92.4}{44} \]

### Step 6: Perform the division.
Now we can simplify \( \frac{92.4}{44} \):
\[ r = 2.1 \text{ cm} \]

### Final Answer:
The radius of the circular ring formed is \( 2.1 \) cm.

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