To find the side length \( x \) of the rhombus formed by gluing the four triangles from the square, we start by recognizing the properties of the triangles.
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Understanding the Triangle Formed: The square has a side length of 4 inches. When the square is cut along its diagonals, it forms four right triangles. Each triangle has legs that are the half-lengths of the square's sides:
- Each leg of the triangle is \( \frac{4}{2} = 2 \) inches.
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Using the Pythagorean Theorem: To find the length of the side of the rhombus, we can consider one of the right triangles, where both legs are known: \[ \text{leg}_1 = 2 \text{ inches} \] \[ \text{leg}_2 = 2 \text{ inches} \] The hypotenuse \( x \) of this right triangle (which will be the side length of the rhombus) can be found using the Pythagorean theorem: \[ x^2 = \text{leg}_1^2 + \text{leg}_2^2 \] \[ x^2 = 2^2 + 2^2 \] \[ x^2 = 4 + 4 = 8 \] \[ x = \sqrt{8} \] \[ x = 2\sqrt{2} \]
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Calculating the Value of \( x \): Now we compute \( 2\sqrt{2} \): \[ 2\sqrt{2} \approx 2 \times 1.4142 \approx 2.8284 \]
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Rounding to Two Decimal Places: Rounding this to two decimal places gives: \[ x \approx 2.83 \text{ inches} \]
Thus, the side length \( x \) of the rhombus is approximately \( \boxed{2.83} \) inches.
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