To solve for the total length of \( BD \) in the rectangle \( BCDA \), we can analyze the segments \( AE \) and \( ED \).
We know:
- \( AE = 3x \)
- \( ED = -4x + 21 \)
In a rectangle, the diagonal \( BD \) is related to the segments \( AE \) and \( ED \) through the relationship between their lengths. Generally, \( BD \) can be calculated as the sum of \( AE \) and \( ED \):
\[ BD = AE + ED \]
Substituting the expressions we have:
\[ BD = 3x + (-4x + 21) \]
Simplifying the equation:
\[ BD = 3x - 4x + 21 \] \[ BD = -x + 21 \]
Now we can analyze the provided responses to identify a possible value for \( x \). To solve for \( x \), we would typically need additional information or constraints, but since none are offered, we can consider that \( BD \) is solvable under conditions where \( -x + 21 \) equals one of the choice responses.
Let’s evaluate each possible response:
- For \( BD = 9 \): \[ -x + 21 = 9 \implies -x = 9 - 21 \implies -x = -12 \implies x = 12 \]
- For \( BD = 3 \): \[ -x + 21 = 3 \implies -x = 3 - 21 \implies -x = -18 \implies x = 18 \]
- For \( BD = 18 \): \[ -x + 21 = 18 \implies -x = 18 - 21 \implies -x = -3 \implies x = 3 \]
- For \( BD = 27 \): \[ -x + 21 = 27 \implies -x = 27 - 21 \implies -x = 6 \implies x = -6 \text{ (not valid)} \]
After checking through the potential values for \( x \), there are valid solutions for \( x = 12, 18, \) and \( 3 \), which correspond to \( BD \) lengths of \( 9, 3, 18 \), respectively.
If you want a total length from the choices given, the only potential response consistent (within reasonable lengths in geometrical contexts) would most likely be:
\( 18 \)
This analysis shows that \( BD = 18 \) is indeed a valid solution for some \( x \) values. Thus, we can select 18 as our answer.