In a triangle `ABC, a =3,c = 5 and b =4, ` then the value of radius of inscribe is `"_____"`

Answer

To find the radius of the inscribed circle (inradius) of triangle ABC with sides \( a = 3 \), \( b = 4 \), and \( c = 5 \), we can use the formula for the inradius \( r \):

\[ r = \frac{\Delta}{s} \]

where \( \Delta \) is the area of the triangle and \( s \) is the semi-perimeter.

### Step 1: Calculate the semi-perimeter \( s \)
The semi-perimeter \( s \) is given by:

\[ s = \frac{a + b + c}{2} \]

Substituting the values:

\[ s = \frac{3 + 4 + 5}{2} = \frac{12}{2} = 6 \]

### Step 2: Calculate the area \( \Delta \) using Heron's formula
Heron's formula states:

\[ \Delta = \sqrt{s(s-a)(s-b)(s-c)} \]

Substituting the values of \( s \), \( a \), \( b \), and \( c \):

\[ \Delta = \sqrt{6(6-3)(6-4)(6-5)} \]
\[ = \sqrt{6 \times 3 \times 2 \times 1} \]
\[ = \sqrt{36} = 6 \]

### Step 3: Calculate the inradius \( r \)
Now that we have both \( \Delta \) and \( s \), we can find the inradius:

\[ r = \frac{\Delta}{s} = \frac{6}{6} = 1 \]

Thus, the radius of the inscribed circle is:

\[ \boxed{1} \]