The in-radius of the triangle whose sides are 3,5,6,is

Options

A. `sqrt(8//7)`
B. `sqrt8`
C. `sqrt7`
D. `sqrt(7//8)`

Answer

To find the in-radius of a triangle with sides 3, 5, and 6, we can follow these steps:

### Step 1: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) of a triangle is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
where \( a, b, c \) are the lengths of the sides of the triangle.

For our triangle:
\[ s = \frac{3 + 5 + 6}{2} = \frac{14}{2} = 7 \]

### Step 2: Calculate the area (Δ) using Heron's formula
Heron's formula for the area \( Δ \) of a triangle is given by:
\[ Δ = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values we have:
\[ Δ = \sqrt{7(7-3)(7-5)(7-6)} \]
Calculating each term:
\[ Δ = \sqrt{7 \times 4 \times 2 \times 1} = \sqrt{56} \]

### Step 3: Calculate the in-radius (r)
The in-radius \( r \) of a triangle can be calculated using the formula:
\[ r = \frac{Δ}{s} \]
Substituting the values we have:
\[ r = \frac{\sqrt{56}}{7} \]

### Step 4: Simplify the in-radius
We can simplify \( \sqrt{56} \):
\[ \sqrt{56} = \sqrt{4 \times 14} = 2\sqrt{14} \]
Thus, we have:
\[ r = \frac{2\sqrt{14}}{7} \]

### Final Answer
The in-radius of the triangle with sides 3, 5, and 6 is:
\[ r = \frac{2\sqrt{14}}{7} \]