If in a triangle with sides a, b & c, `(s-a)=5cm,(s-b)=10cm&(s-c)=1cm` find area of the triangle.

Answer

To find the area of the triangle given the conditions \(s-a=5 \, \text{cm}\), \(s-b=10 \, \text{cm}\), and \(s-c=1 \, \text{cm}\), we can follow these steps:

### Step 1: Express \(a\), \(b\), and \(c\) in terms of \(s\)
From the given equations:
- \(s - a = 5\) implies \(a = s - 5\)
- \(s - b = 10\) implies \(b = s - 10\)
- \(s - c = 1\) implies \(c = s - 1\)

### Step 2: Find the semi-perimeter \(s\)
The semi-perimeter \(s\) is defined as:
\[ s = \frac{a + b + c}{2} \]
Substituting the expressions for \(a\), \(b\), and \(c\):
\[ s = \frac{(s - 5) + (s - 10) + (s - 1)}{2} \]
This simplifies to:
\[ s = \frac{3s - 16}{2} \]
Multiplying both sides by 2:
\[ 2s = 3s - 16 \]
Rearranging gives:
\[ s = 16 \, \text{cm} \]

### Step 3: Calculate \(a\), \(b\), and \(c\)
Now substituting \(s = 16\) back into the expressions for \(a\), \(b\), and \(c\):
- \(a = s - 5 = 16 - 5 = 11 \, \text{cm}\)
- \(b = s - 10 = 16 - 10 = 6 \, \text{cm}\)
- \(c = s - 1 = 16 - 1 = 15 \, \text{cm}\)

### Step 4: Use Heron's formula to find the area
Heron's formula states that the area \(A\) of a triangle can be calculated as:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values we have:
\[ A = \sqrt{16 \times (16 - 11) \times (16 - 6) \times (16 - 15)} \]
Calculating each term:
- \(s - a = 5\)
- \(s - b = 10\)
- \(s - c = 1\)

Thus, we have:
\[ A = \sqrt{16 \times 5 \times 10 \times 1} \]
Calculating the product:
\[ A = \sqrt{800} \]
Breaking it down:
\[ 800 = 16 \times 50 = 16 \times 25 \times 2 \]
Taking the square root:
\[ A = \sqrt{16} \times \sqrt{25} \times \sqrt{2} = 4 \times 5 \times \sqrt{2} = 20\sqrt{2} \, \text{cm}^2 \]

### Final Answer
The area of the triangle is:
\[ \boxed{20\sqrt{2} \, \text{cm}^2} \]