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

Answer

To solve the problem, we need to find the sides \( a \), \( b \), and \( c \) of the triangle given the conditions \( (s-a) = 5 \, \text{cm} \), \( (s-b) = 10 \, \text{cm} \), and \( (s-c) = 1 \, \text{cm} \).

### Step-by-Step Solution:

1. **Understanding the Semi-Perimeter**:
The semi-perimeter \( s \) of a triangle is defined as:
\[ s = \frac{a + b + c}{2} \]
This means:
\[ a + b + c = 2s \]

2. **Setting Up the Equations**:
From the given conditions, we can express \( a \), \( b \), and \( c \) in terms of \( s \):
- From \( s - a = 5 \):
\[ a = s - 5 \]
- From \( s - b = 10 \):
\[ b = s - 10 \]
- From \( s - c = 1 \):
\[ c = s - 1 \]

3. **Adding the Equations**:
Now, we add the three equations:
\[ (s - a) + (s - b) + (s - c) = 5 + 10 + 1 \]
This simplifies to:
\[ 3s - (a + b + c) = 16 \]

4. **Substituting for \( a + b + c \)**:
We know from the semi-perimeter definition that \( a + b + c = 2s \). Substituting this into the equation gives:
\[ 3s - 2s = 16 \]
Therefore:
\[ s = 16 \, \text{cm} \]

5. **Finding the Values of \( a \), \( b \), and \( c \)**:
Now that we have \( s \), we can find \( a \), \( b \), and \( c \):
- For \( a \):
\[ a = s - 5 = 16 - 5 = 11 \, \text{cm} \]
- For \( b \):
\[ b = s - 10 = 16 - 10 = 6 \, \text{cm} \]
- For \( c \):
\[ c = s - 1 = 16 - 1 = 15 \, \text{cm} \]

### Final Values:
Thus, the sides of the triangle are:
- \( a = 11 \, \text{cm} \)
- \( b = 6 \, \text{cm} \)
- \( c = 15 \, \text{cm} \)