To find the length of the diagonal of the rectangle formed by the points A \((1, 2)\), B \((3, 2)\), C \((3, 4)\), and D \((1, 4)\), we can use the Pythagorean Theorem.
First, we can determine the width and height of the rectangle:
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Width (from A to B or D to C): \[ x_B - x_A = 3 - 1 = 2 \]
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Height (from A to D or B to C): \[ y_C - y_B = 4 - 2 = 2 \]
Now, we can apply the Pythagorean Theorem, which states: \[ c = \sqrt{a^2 + b^2} \] where \(a\) and \(b\) are the sides of the rectangle, and \(c\) is the diagonal.
Substituting the width and height into the equation: \[ c = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
Now we can approximate \(2\sqrt{2}\): \[ \sqrt{2} \approx 1.414 \Rightarrow 2\sqrt{2} \approx 2 \times 1.414 \approx 2.828 \]
Rounding this to the nearest hundredth gives us approximately \(2.83\).
Thus, the length of the diagonal rounded to the nearest hundredth is:
2.83