To solve for the total length of \( BD \) in rectangle \( ABCD \), we must first analyze the given information about the sides \( ARE \) and \( ED \).
We're given:
- \( ARE = 3x \)
- \( ED = -4x + 21 \)
In a rectangle, opposite sides are equal. This implies:
- \( AB = CD \) (one pair of opposite sides)
- \( AD = BC \) (the other pair of opposite sides)
However, we don't have the lengths of \( AB \) and \( AD \) directly. Instead, we can infer the situation from the lengths provided.
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In order to equate \( ARE \) and \( ED \), we can set them equal to find \( x \): \[ 3x = -4x + 21 \] Rearranging gives: \[ 3x + 4x = 21 \] \[ 7x = 21 \] \[ x = 3 \]
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Now, substitute \( x \) back into the equations for \( ARE \) and \( ED \):
- \( ARE = 3(3) = 9 \)
- \( ED = -4(3) + 21 = -12 + 21 = 9 \)
Since both \( ARE \) and \( ED \) equal 9, we can infer that the lengths of \( BD \) (which can also be a diagonal) of the rectangle can be calculated using the Pythagorean theorem if needed. However, since both side lengths measure equally (9), we check the options available which correspond to:
Input Choices: \( 3, 3, 9, 9, 27, 27, 18 \)
Identify the consistent value:
- The lengths provided indicate that \( BD = 9 \).
Thus, the total length of \( BD \) is \( 9 \).
Final Answer: 9
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