To find the total length of \( BD \) in the rectangle \( BCDA \) with segments \( AE = 3x \) and \( ED = -4x + 21 \), we start by recognizing that \( BD \) can be calculated using the segments \( AE \) and \( ED \).
Since points \( B \), \( C \), \( D \), and \( A \) are the vertices of rectangle \( BCDA \), the sides opposite each other are equal, and \( BD \) is the diagonal.
Since \( E \) is the point on diagonal \( BD \) that splits it into two segments \( AE \) and \( ED \), we have: \[ BD = AE + ED \]
Substituting the expressions we have: \[ BD = 3x + (-4x + 21) \] Simplifying this gives: \[ BD = 3x - 4x + 21 = -x + 21 \]
To find a specific value for \( BD \), we need to determine \( x \). However, without a specific value or further constraints on \( x \), we can push forward by analyzing the equation:
- Rearrange to find \( x \): \[ x = 21 - BD \]
This equation indicates that the length of \( BD \) depends on the value we choose for \( x \). Testing possible values:
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If we were to set \( BD = 9 \): \[ BD = 9 \Rightarrow x = 21 - 9 = 12 \]
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If we set \( BD = 3 \): \[ BD = 3 \Rightarrow x = 21 - 3 = 18 \]
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If \( BD = 18 \): \[ BD = 18 \Rightarrow x = 21 - 18 = 3 \]
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If \( BD = 27 \): \[ BD = 27 \Rightarrow x = 21 - 27 = -6 \]
This means we have valid values of \( x \) for \( BD = 9, 3, \text{ or } 18 \), but not for \( BD = 27 \) since \( x \) must remain positive. The nature of the problem suggests that some arbitrary values of \( BD \) might not yield coherent dimensions based on \( AE \) and \( ED \).
Upon testing the possible options:
- \( BD = 9 \) leads to feasible positive dimensions.
- \( BD = 3 \) also leads to feasible positive dimensions.
- \( BD = 18 \) results in feasible dimensions.
Thus, while the correct total length of \( BD \) with the values provided appears variable, the option that fits best and encompasses coherence in terms of \( x > 0 \) is 3, 9, or 18. But if the question seeks the maximum total feasible length compliant with the structure given the variables, we highlight \( 18 \).
Thus the most suitable answer from the options would be: \[ \text{C. } 18 \] as it yields \( x \) as positive without complications or nonsensical outputs.
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