To find the total length of \( BD \) in the rectangle \( BCDA \), we can use the information given about the lengths of segments \( AE \) and \( ED \).
We know:
- \( AE = 3x \)
- \( ED = -4x + 21 \)
The total length of segment \( BD \) can be expressed as:
\[ BD = AE + ED = 3x + (-4x + 21) = 3x - 4x + 21 = -x + 21 \]
To complete the problem, we need to determine a specific value for \( x \) so we can find \( BD \). However, based on the options provided (3, 27, 9, 18), we should test these values to see if they fit within realistic constraints for the length of the segment.
Let’s solve for \( x \) when \( BD \) equals each possible option:
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For \( BD = 3 \): \[ -x + 21 = 3 \Rightarrow -x = 3 - 21 \Rightarrow -x = -18 \Rightarrow x = 18 \]
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For \( BD = 27 \): \[ -x + 21 = 27 \Rightarrow -x = 27 - 21 \Rightarrow -x = 6 \Rightarrow x = -6 \]
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For \( BD = 9 \): \[ -x + 21 = 9 \Rightarrow -x = 9 - 21 \Rightarrow -x = -12 \Rightarrow x = 12 \]
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For \( BD = 18 \): \[ -x + 21 = 18 \Rightarrow -x = 18 - 21 \Rightarrow -x = -3 \Rightarrow x = 3 \]
Now, we have found \( x \) values resulting from setting \( BD \) to each of the potential lengths.
To determine which of these lengths is valid, we should ensure that the computed \( AE \) and \( ED \) lengths yield positive values.
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For \( x = 18 \) (leading to \( BD = 3 \)):
- \( AE = 3(18) = 54 \)
- \( ED = -4(18) + 21 = -72 + 21 = -51 \) (not valid)
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For \( x = -6 \) (leading to \( BD = 27 \)):
- \( AE = 3(-6) = -18 \) (not valid)
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For \( x = 12 \) (leading to \( BD = 9 \)):
- \( AE = 3(12) = 36 \)
- \( ED = -4(12) + 21 = -48 + 21 = -27 \) (not valid)
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For \( x = 3 \) (leading to \( BD = 18 \)):
- \( AE = 3(3) = 9 \)
- \( ED = -4(3) + 21 = -12 + 21 = 9 \) (both positive).
Thus, the valid total length of \( BD \) is:
\[ \boxed{18} \]
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