To determine the value of \( LK \) that would make the quadrilateral \( LMKF \) a rhombus, we start by noting that in a rhombus, all four sides must be of equal length.
We have:
- \( LA = -3x + 4 \)
- \( LK = -5x \)
For \( LMKF \) to be a rhombus, the lengths of \( LA \) and \( LK \) must be equal, thus we can set them equal to each other:
\[ -3x + 4 = -5x \]
Now, let's solve for \( x \):
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Rearranging the equation gives: \[ -3x + 5x = -4 \] \[ 2x = -4 \]
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Dividing both sides by 2 gives: \[ x = -2 \]
Next, we substitute \( x = -2 \) back into the expressions for \( LA \) and \( LK \):
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For \( LA \): \[ LA = -3(-2) + 4 = 6 + 4 = 10 \]
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For \( LK \): \[ LK = -5(-2) = 10 \]
Thus, both \( LA \) and \( LK \) are equal to 10. Therefore, the value of \( LK \) required for \( LMKF \) to be a rhombus is:
10