Find the ex-radius of a ex-circle which touches side BC.
`AB=13 cm, BC=14 cm, CA=15cm`.

Answer

To find the ex-radius of the ex-circle that touches side BC of triangle ABC, we can follow these steps:

### Step 1: Identify the sides of the triangle
Given:
- AB = 13 cm
- BC = 14 cm
- CA = 15 cm

### Step 2: Calculate the semi-perimeter (s) of the triangle
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{AB + BC + CA}{2} \]
Substituting the values:
\[ s = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 \text{ cm} \]

### Step 3: Calculate the area of the triangle using Heron's formula
Heron's formula states that the area \( A \) of a triangle can be calculated as:
\[ A = \sqrt{s(s - a)(s - b)(s - c)} \]
where \( a, b, c \) are the lengths of the sides of the triangle. Here, we can take:
- \( a = AB = 13 \text{ cm} \)
- \( b = BC = 14 \text{ cm} \)
- \( c = CA = 15 \text{ cm} \)

Now substituting the values into Heron's formula:
\[ A = \sqrt{21 \times (21 - 13) \times (21 - 14) \times (21 - 15)} \]
Calculating each term:
\[ A = \sqrt{21 \times 8 \times 7 \times 6} \]

### Step 4: Simplify the expression
Calculating the product:
\[ 21 \times 8 = 168 \]
\[ 168 \times 7 = 1176 \]
\[ 1176 \times 6 = 7056 \]
So,
\[ A = \sqrt{7056} \]

### Step 5: Calculate the square root
To find \( \sqrt{7056} \), we can factor it:
\[ 7056 = 2^4 \times 3^2 \times 7^2 \]
Taking the square root:
\[ \sqrt{7056} = 2^2 \times 3 \times 7 = 4 \times 3 \times 7 = 84 \text{ cm}^2 \]

### Step 6: Calculate the ex-radius (r_a) of the ex-circle touching side BC
The ex-radius \( r_a \) is given by the formula:
\[ r_a = \frac{A}{s - a} \]
Here, \( a = BC = 14 \text{ cm} \).
Thus:
\[ r_a = \frac{84}{21 - 14} = \frac{84}{7} = 12 \text{ cm} \]

### Final Answer
The ex-radius of the ex-circle that touches side BC is **12 cm**.

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