The sides of a triangle are 13,14,15, then the radius of its in-circle is

Options

A. `67//8`
B. `65//4`
C. 4
D. 24

Answer

To find the radius of the in-circle of a triangle with sides 13, 14, and 15, we can follow these steps:

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

For our triangle:
\[ s = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 \]

### Step 2: Calculate the Area (Δ) using Heron's Formula
Heron's formula for the area \( Δ \) of a triangle is given by:
\[ Δ = \sqrt{s(s-a)(s-b)(s-c)} \]
where \( a, b, c \) are the lengths of the sides and \( s \) is the semi-perimeter.

Substituting the values:
\[ Δ = \sqrt{21 \times (21 - 13) \times (21 - 14) \times (21 - 15)} \]
Calculating each term:
\[ Δ = \sqrt{21 \times 8 \times 7 \times 6} \]

### Step 3: Simplify the Area Calculation
Now we simplify:
\[ Δ = \sqrt{21 \times 8 \times 7 \times 6} \]
Breaking it down:
\[ = \sqrt{21 \times 7 \times 48} = \sqrt{147 \times 48} \]
Now, simplifying further:
\[ = \sqrt{147 \times 16 \times 3} = \sqrt{2352} \]
Factoring out:
\[ = \sqrt{(7^2) \times (3^2) \times (2^4)} = 84 \]

### Step 4: Calculate the Inradius (r)
The radius \( r \) of the in-circle is given by the formula:
\[ r = \frac{Δ}{s} \]
Substituting the values we have:
\[ r = \frac{84}{21} = 4 \]

### Conclusion
Thus, the radius of the in-circle is \( r = 4 \).

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