The sides of a triangle are 8 cm, 15 cm and 17 cm. The sum of radii of circumcircle and incircle of the triangle is

Options

A. 23 cm
B. 11.5 cm
C. 25 cm
D. 12.5 cm

Answer

To find the sum of the radii of the circumcircle and incircle of a triangle with sides 8 cm, 15 cm, and 17 cm, we can follow these steps:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( a = 8 \) cm
- \( b = 15 \) cm
- \( c = 17 \) cm

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{8 + 15 + 17}{2} = \frac{40}{2} = 20 \text{ cm} \]

### Step 3: Calculate the area (A) of the triangle
We can use Heron's formula to find the area:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values:
\[ A = \sqrt{20(20-8)(20-15)(20-17)} = \sqrt{20 \times 12 \times 5 \times 3} \]
Calculating inside the square root:
\[ A = \sqrt{20 \times 12 \times 5 \times 3} = \sqrt{3600} = 60 \text{ cm}^2 \]

### Step 4: Calculate the circumradius (R)
The circumradius \( R \) is given by the formula:
\[ R = \frac{abc}{4A} \]
Substituting the values:
\[ R = \frac{8 \times 15 \times 17}{4 \times 60} \]
Calculating the numerator:
\[ 8 \times 15 = 120 \quad \text{and} \quad 120 \times 17 = 2040 \]
Now substituting back:
\[ R = \frac{2040}{240} = \frac{34}{4} = 8.5 \text{ cm} \]

### Step 5: Calculate the inradius (r)
The inradius \( r \) is given by the formula:
\[ r = \frac{A}{s} \]
Substituting the values:
\[ r = \frac{60}{20} = 3 \text{ cm} \]

### Step 6: Calculate the sum of the circumradius and inradius
Now, we add the circumradius and inradius:
\[ R + r = 8.5 + 3 = 11.5 \text{ cm} \]

### Final Answer
The sum of the radii of the circumcircle and incircle of the triangle is:
\[ \boxed{11.5 \text{ cm}} \]