`(a)/(b) = (c)/(d) = (e)/(f) = k(a, b, c, d, e, f gt 0)` then (A) `k = (a + c +e)/(b + d + f)` (B) `k = (a^(2) + c^(2) +e^(2))/(b^(2) + d^(2) + f^(2))` (C) `k = (c^(2)e^(2))/(d^(2)f^(2))` (D) `k = ((ace)/(bdf))^((1)/(3))`

Options

A. `k = (a + c +e)/(b + d + f)`
B. `k = (a^(2) + c^(2) +e^(2))/(b^(2) + d^(2) + f^(2))`
C. `k = (c^(2)e^(2))/(d^(2)f^(2))`
D. `k = ((ace)/(bdf))^((1)/(3))`

Answer

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

\[ \frac{a}{b} = \frac{c}{d} = \frac{e}{f} = k \]

From this, we can express \(a\), \(c\), and \(e\) in terms of \(k\), \(b\), \(d\), and \(f\):

1. **Express \(a\), \(c\), and \(e\) in terms of \(k\)**:
- \(a = k \cdot b\)
- \(c = k \cdot d\)
- \(e = k \cdot f\)

Now, we will check each option one by one.

### Option A:
\[ k = \frac{a + c + e}{b + d + f} \]

Substituting the values of \(a\), \(c\), and \(e\):

\[ k = \frac{k \cdot b + k \cdot d + k \cdot f}{b + d + f} \]

Factoring \(k\) out from the numerator:

\[ k = \frac{k(b + d + f)}{b + d + f} \]

Since \(b + d + f \neq 0\) (as \(b, d, f > 0\)), we can cancel \(b + d + f\):

\[ k = k \]

This is true, so **Option A is correct**.

### Option B:
\[ k = \frac{a^2 + c^2 + e^2}{b^2 + d^2 + f^2} \]

Substituting the values:

\[ k = \frac{(k \cdot b)^2 + (k \cdot d)^2 + (k \cdot f)^2}{b^2 + d^2 + f^2} \]

This simplifies to:

\[ k = \frac{k^2(b^2 + d^2 + f^2)}{b^2 + d^2 + f^2} \]

Again, since \(b^2 + d^2 + f^2 \neq 0\):

\[ k = k^2 \]

This implies \(k^2 - k = 0\) or \(k(k - 1) = 0\). Since \(k > 0\), this means \(k \neq 1\) is not guaranteed. Thus, **Option B is incorrect**.

### Option C:
\[ k = \frac{c^2 e^2}{d^2 f^2} \]

Substituting the values:

\[ k = \frac{(k \cdot d)^2 (k \cdot f)^2}{d^2 f^2} \]

This simplifies to:

\[ k = \frac{k^2 d^2 k^2 f^2}{d^2 f^2} = \frac{k^4 d^2 f^2}{d^2 f^2} \]

Again, since \(d^2 f^2 \neq 0\):

\[ k = k^4 \]

This implies \(k^4 - k = 0\) or \(k(k^3 - 1) = 0\). Since \(k > 0\), this means \(k \neq 1\) is not guaranteed. Thus, **Option C is incorrect**.

### Option D:
\[ k = \left(\frac{a c e}{b d f}\right)^{\frac{1}{3}} \]

Substituting the values:

\[ k = \left(\frac{(k \cdot b)(k \cdot d)(k \cdot f)}{b d f}\right)^{\frac{1}{3}} \]

This simplifies to:

\[ k = \left(\frac{k^3 b d f}{b d f}\right)^{\frac{1}{3}} = \left(k^3\right)^{\frac{1}{3}} = k \]

This is true, so **Option D is correct**.

### Conclusion:
The correct options are **A** and **D**.

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