To find the side length \( x \) of the rhombus formed by gluing the four equal triangles cut from a square with a side length of 4 inches, we start by determining the properties of the square and the triangles.
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Identify the triangles: The square has a side length of 4 inches. When we cut along the diagonals, we divide the square into four right-angled triangles. Each triangle will have legs measuring 4 inches (the sides of the square).
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Calculate the hypotenuse of one triangle: In each triangle, both legs are 4 inches. We can use the Pythagorean theorem to determine the hypotenuse \( c \): \[ c = \sqrt{a^2 + b^2} \] where \( a \) and \( b \) are the legs of the triangle. Here, \( a = 4 \) and \( b = 4 \): \[ c = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Calculate the side length of the rhombus: The side length \( x \) of the rhombus is equal to the hypotenuse of the triangles. Therefore, we have: \[ x = 4\sqrt{2} \]
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Find the numerical value of \( x \): To find the decimal value, we calculate \( 4\sqrt{2} \): \[ \sqrt{2} \approx 1.41421 \implies 4\sqrt{2} \approx 4 \times 1.41421 \approx 5.65684 \]
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Round to two decimal places: Rounding \( 5.65684 \) to two decimal places gives us: \[ x \approx 5.66 \]
Thus, the side length of the rhombus is approximately \( \boxed{5.66} \) inches.
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