If `A/3 = B/2 = C/5`, then what is the valuee of ratio `(C+A)^(2) : (A + B)^(2): (B+C)^(2)`?

Options

A. `9 : 4 : 25 `
B. `25 : 4 : 9`
C. `64 : 25 : 49`
D. `49 : 25 : 64`

Answer

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

\[ \frac{A}{3} = \frac{B}{2} = \frac{C}{5} \]

Let's denote the common ratio as \( k \). Therefore, we can express \( A \), \( B \), and \( C \) in terms of \( k \):

1. \( A = 3k \)
2. \( B = 2k \)
3. \( C = 5k \)

Now we need to find the values of \( C + A \), \( A + B \), and \( B + C \):

1. **Calculate \( C + A \)**:
\[ C + A = 5k + 3k = 8k \]

2. **Calculate \( A + B \)**:
\[ A + B = 3k + 2k = 5k \]

3. **Calculate \( B + C \)**:
\[ B + C = 2k + 5k = 7k \]

Next, we need to square these sums to find the required ratio:

1. **Calculate \( (C + A)^2 \)**:
\[ (C + A)^2 = (8k)^2 = 64k^2 \]

2. **Calculate \( (A + B)^2 \)**:
\[ (A + B)^2 = (5k)^2 = 25k^2 \]

3. **Calculate \( (B + C)^2 \)**:
\[ (B + C)^2 = (7k)^2 = 49k^2 \]

Now we can write the ratio:

\[ (C + A)^2 : (A + B)^2 : (B + C)^2 = 64k^2 : 25k^2 : 49k^2 \]

Since \( k^2 \) is common in all terms, we can simplify the ratio:

\[ 64 : 25 : 49 \]

Thus, the final answer for the ratio \((C+A)^2 : (A+B)^2 : (B+C)^2\) is:

\[ \boxed{64 : 25 : 49} \]