To find the value of \( x \), we first need to express the perimeters of both shapes in terms of \( x \).
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Perimeter of the square: The side length of the square is \( x + 5 \) cm. The perimeter \( P_{\text{square}} \) of a square is given by the formula: \[ P_{\text{square}} = 4 \times \text{side length} = 4(x + 5) \] Expanding this, we get: \[ P_{\text{square}} = 4x + 20 \]
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Perimeter of the regular hexagon: The side length of the hexagon is \( x - 1 \) cm. The perimeter \( P_{\text{hexagon}} \) of a regular hexagon is: \[ P_{\text{hexagon}} = 6 \times \text{side length} = 6(x - 1) \] Expanding this gives: \[ P_{\text{hexagon}} = 6x - 6 \]
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Setting the perimeters equal: Since the problem states that the two figures have the same perimeter, we set the equations for their perimeters equal to each other: \[ 4x + 20 = 6x - 6 \]
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Solving for \( x \): We can solve this equation step by step: \[ 4x + 20 = 6x - 6 \] Subtract \( 4x \) from both sides: \[ 20 = 2x - 6 \] Add \( 6 \) to both sides: \[ 26 = 2x \] Finally, divide both sides by \( 2 \): \[ x = 13 \]
So, the value of \( x \) is \( \boxed{13} \).
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