To find the value of \( x \) given that the perimeters of Raji's square and Ava's hexagon are the same, we can start by calculating the perimeters of both shapes.
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Perimeter of the square: The perimeter \( P \) of a square with side length \( s \) is given by the formula: \[ P = 4s \] For Raji's square with side length \( x + 5 \): \[ P_{square} = 4(x + 5) \]
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Perimeter of the hexagon: The perimeter \( P \) of a regular hexagon with side length \( s \) is given by the formula: \[ P = 6s \] For Ava's hexagon with side length \( x - 1 \): \[ P_{hexagon} = 6(x - 1) \]
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Set the perimeters equal to each other: Since the perimeters are equal, we have: \[ 4(x + 5) = 6(x - 1) \]
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Solve for \( x \): Expanding both sides: \[ 4x + 20 = 6x - 6 \] Rearranging gives: \[ 20 + 6 = 6x - 4x \] This simplifies to: \[ 26 = 2x \] Dividing both sides by 2: \[ x = 13 \]
Thus, the solution is \( x = 13 \).