In a triangle `ABC, (s-a): (s-b) : (s -c) ::11 : 8: 7` where s is semi perimeter a,b,c are sides
find the ratio of a:b:c

Options

A. `7:8:11`
B. `14:17:19`
C. `15:18:19`
D. `9:13:21`

Answer

To solve the problem step by step, we will use the information given about the triangle and the ratios involving the semi-perimeter.

### Step 1: Define the semi-perimeter and the sides of the triangle
Let \( s \) be the semi-perimeter of triangle \( ABC \), and let the sides opposite to vertices \( A, B, C \) be \( a, b, c \) respectively.

Given the ratios:
\[ (s-a):(s-b):(s-c) = 11:8:7 \]

### Step 2: Express \( s-a, s-b, s-c \) in terms of a common variable \( K \)
From the given ratios, we can express:
\[ s - a = 11K, \quad s - b = 8K, \quad s - c = 7K \]

### Step 3: Add the equations
Now, we can add these three equations:
\[ (s-a) + (s-b) + (s-c) = 11K + 8K + 7K \]
This simplifies to:
\[ 3s - (a + b + c) = 26K \]

### Step 4: Relate semi-perimeter to the sides
We know that the semi-perimeter \( s \) is defined as:
\[ s = \frac{a + b + c}{2} \]
Thus, we can substitute \( a + b + c \) in terms of \( s \):
\[ 3s - 2s = 26K \quad \Rightarrow \quad s = 26K \]

### Step 5: Substitute \( s \) back to find \( a, b, c \)
Now we can substitute \( s \) back into the equations for \( a, b, c \):
1. For \( a \):
\[ s - a = 11K \quad \Rightarrow \quad 26K - a = 11K \quad \Rightarrow \quad a = 26K - 11K = 15K \]
2. For \( b \):
\[ s - b = 8K \quad \Rightarrow \quad 26K - b = 8K \quad \Rightarrow \quad b = 26K - 8K = 18K \]
3. For \( c \):
\[ s - c = 7K \quad \Rightarrow \quad 26K - c = 7K \quad \Rightarrow \quad c = 26K - 7K = 19K \]

### Step 6: Write the ratio of the sides
Now we have:
\[ a = 15K, \quad b = 18K, \quad c = 19K \]
Thus, the ratio \( a:b:c \) is:
\[ a:b:c = 15K:18K:19K \quad \Rightarrow \quad 15:18:19 \]

### Final Answer
The ratio of the sides \( a:b:c \) is \( 15:18:19 \).

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