To find the total length of \( BD \) in rectangle \( BCDA \) given that \( AE \) can be expressed as \( 3x \) and \( ED \) as \( -4x + 21 \), we can analyze the dimensions of the rectangle.
In rectangle \( BCDA \):
- \( AE \) and \( ED \) are segments on the diagonal \( BD \).
- The entire length of diagonal \( BD \) is the sum of the lengths of segments \( AE \) and \( ED \).
Therefore, we add the two expressions together:
\[ BD = AE + ED = 3x + (-4x + 21) \]
Simplifying this expression:
\[ BD = 3x - 4x + 21 = -x + 21 \]
Thus, the total length of \( BD \) is given by \( -x + 21 \).
This leads us to conclude that the length of diagonal \( BD \) is dependent on the value of \( x \).
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