To find the value of \( x \) where the perimeter of the square and the perimeter of the hexagon are equal, we will first calculate the perimeters of both shapes.
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Perimeter of the square: A square with side length \( x + 5 \) cm has a perimeter given by the formula: \[ \text{Perimeter of square} = 4 \times (\text{side length}) = 4 \times (x + 5) = 4x + 20 \text{ cm} \]
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Perimeter of the hexagon: A regular hexagon with side length \( x - 1 \) cm has a perimeter given by the formula: \[ \text{Perimeter of hexagon} = 6 \times (\text{side length}) = 6 \times (x - 1) = 6x - 6 \text{ cm} \]
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Set the perimeters equal: Since the problem states that the two figures have the same perimeter, we can set the two perimeter expressions equal to each other: \[ 4x + 20 = 6x - 6 \]
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Solve for \( x \): To solve for \( x \), we will rearrange the equation: \[ 4x + 20 = 6x - 6 \] Subtract \( 4x \) from both sides: \[ 20 = 6x - 4x - 6 \] This simplifies to: \[ 20 = 2x - 6 \] Next, add \( 6 \) to both sides: \[ 20 + 6 = 2x \] \[ 26 = 2x \] Finally, divide by \( 2 \): \[ x = \frac{26}{2} = 13 \]
Thus, the value of \( x \) is \( \boxed{13} \).
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