The sides of a triangle are 11 cm, 60 cm and 61 cm. What is the radius of the circile circumscribing the triangle ?

Options

A. 31 cm
B. 30 cm
C. 30.5 cm
D. 31.5 cm

Answer

To find the radius of the circle circumscribing the triangle with sides 11 cm, 60 cm, and 61 cm, we can follow these steps:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( a = 11 \) cm
- \( b = 60 \) cm
- \( c = 61 \) cm

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) of the triangle is given by the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{11 + 60 + 61}{2} = \frac{132}{2} = 66 \text{ cm} \]

### Step 3: Calculate the area of the triangle (A)
Using Heron's formula, the area \( A \) of the triangle can be calculated as:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values:
\[ A = \sqrt{66 \times (66 - 11) \times (66 - 60) \times (66 - 61)} \]
Calculating each term:
- \( s - a = 66 - 11 = 55 \)
- \( s - b = 66 - 60 = 6 \)
- \( s - c = 66 - 61 = 5 \)

Now substituting these values into the area formula:
\[ A = \sqrt{66 \times 55 \times 6 \times 5} \]
Calculating the product:
\[ 66 \times 55 = 3630 \]
\[ 6 \times 5 = 30 \]
Now:
\[ A = \sqrt{3630 \times 30} = \sqrt{108900} = 330 \text{ cm}^2 \]

### Step 4: Calculate the radius of the circumscribing circle (R)
The radius \( R \) of the circumscribing circle is given by the formula:
\[ R = \frac{abc}{4A} \]
Substituting the values:
\[ R = \frac{11 \times 60 \times 61}{4 \times 330} \]
Calculating the numerator:
\[ 11 \times 60 = 660 \]
\[ 660 \times 61 = 40260 \]
Now substituting back into the formula:
\[ R = \frac{40260}{1320} = 30.5 \text{ cm} \]

### Final Answer
The radius of the circle circumscribing the triangle is \( 30.5 \) cm.

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