To find the side length \( x \) of the rhombus formed by the four triangles cut from the square, we can apply the properties of the square and the Pythagorean Theorem.
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Determine the Area of the Square: The area of the square is given by side length squared. Here the side length is 4 inches. \[ \text{Area of square} = 4 \times 4 = 16 \text{ square inches} \]
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Understand the Geometry of the Triangles: The two diagonals of the square intersect at their midpoints, dividing the square into four equal triangles. Each triangle is a right triangle, and the legs of these triangles are the halves of the sides of the square.
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Calculate the Lengths of the Legs of the Triangles: The length of each leg is half the side of the square: \[ \text{Length of each leg} = \frac{4}{2} = 2 \text{ inches} \]
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Applying the Pythagorean Theorem: Each triangle has legs of length 2 inches. The hypotenuse \( x \) (the side of the rhombus) is given by the Pythagorean Theorem: \[ x = \sqrt{(\text{leg})^2 + (\text{leg})^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} \]
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Simplify \( \sqrt{8} \): \[ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \]
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Calculate \( 2\sqrt{2} \) using a calculator:
- Using \(\sqrt{2} \approx 1.414\), \[ 2\sqrt{2} \approx 2 \times 1.414 = 2.828 \]
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Round to Two Decimal Places: \[ x \approx 2.83 \text{ inches} \]
Thus, the side length of the rhombus \( x \) is approximately: \[ \boxed{2.83} \text{ inches} \]
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