To find the total length of BD in rectangle BCDA with sides AE and ED, we can sum the lengths of AE and ED.
Let's express the total length BD:
- AE = 3x
- ED = -4x + 21
We need to find the total length of BD, which is the sum of AE and ED:
\[ BD = AE + ED = 3x + (-4x + 21) \]
Now simplify the expression:
\[ BD = 3x - 4x + 21 \] \[ BD = -x + 21 \]
To determine the potential lengths given as options (A. 27, B. 18, C. 3, D. 9), we can set \( -x + 21 \) equal to each of the options and solve for \( x \):
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Option A: \( -x + 21 = 27 \) \[ -x = 27 - 21 \] \[ -x = 6 \quad \Rightarrow \quad x = -6 \]
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Option B: \( -x + 21 = 18 \) \[ -x = 18 - 21 \] \[ -x = -3 \quad \Rightarrow \quad x = 3 \]
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Option C: \( -x + 21 = 3 \) \[ -x = 3 - 21 \] \[ -x = -18 \quad \Rightarrow \quad x = 18 \]
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Option D: \( -x + 21 = 9 \) \[ -x = 9 - 21 \] \[ -x = -12 \quad \Rightarrow \quad x = 12 \]
Now we can check the possible \( x \) values just calculated, but since none of the resulting \( x \) values correlate directly with the options we started with, we should focus on determining if specific values of \( x \) yield integer lengths for BD.
- If \( x = 3 \): \[ BD = -3 + 21 = 18 \] (Option B)
Thus, the value of \( BD \) that fits the options is B. 18.
Therefore, the correct answer is:
B. 18.
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