`xy : yz : zx=2 : 3 :7`
Find `(x+y) : (y+z) : (z+x)=?`

Answer

To solve the problem, we start with the given ratio:

**Given:**
\[ xy : yz : zx = 2 : 3 : 7 \]

### Step 1: Assign Variables
Let us assume a common variable \( k \) such that:
- \( xy = 2k \)
- \( yz = 3k \)
- \( zx = 7k \)

### Step 2: Express Each Variable
From the equations, we can express \( x, y, z \) in terms of \( k \).

1. From \( xy = 2k \):
\[ y = \frac{2k}{x} \]

2. From \( yz = 3k \):
\[ z = \frac{3k}{y} \]

3. From \( zx = 7k \):
\[ z = \frac{7k}{x} \]

### Step 3: Finding Ratios
Now we will find the ratios of \( x, y, z \).

1. From \( xy \) and \( yz \):
\[ \frac{xy}{yz} = \frac{2k}{3k} \implies \frac{x}{z} = \frac{2}{3} \implies x : z = 2 : 3 \]

2. From \( yz \) and \( zx \):
\[ \frac{yz}{zx} = \frac{3k}{7k} \implies \frac{y}{x} = \frac{3}{7} \implies y : x = 3 : 7 \]

### Step 4: Finding a Common Ratio
Now we have two ratios:
- \( x : z = 2 : 3 \)
- \( y : x = 3 : 7 \)

To combine these ratios, we need to express them with a common value for \( x \).

Let’s scale the first ratio \( x : z = 2 : 3 \) by 7 (the value of \( x \) in the second ratio) and the second ratio \( y : x = 3 : 7 \) by 2 (the value of \( x \) in the first ratio):
- \( x : z = 14 : 21 \)
- \( y : x = 6 : 14 \)

### Step 5: Combine Ratios
Now we can write:
- \( x : y : z = 14 : 6 : 21 \)

### Step 6: Calculate \( (x+y) : (y+z) : (z+x) \)
Now we calculate:
1. \( x + y = 14 + 6 = 20 \)
2. \( y + z = 6 + 21 = 27 \)
3. \( z + x = 21 + 14 = 35 \)

Thus, we have:
\[ (x+y) : (y+z) : (z+x) = 20 : 27 : 35 \]

### Final Answer:
\[ \boxed{20 : 27 : 35} \]