To find the length between the points (2, 5) and (7, 3) using the Pythagorean Theorem, we first determine the differences in the x-coordinates and the y-coordinates.
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Calculate the difference in x-coordinates: \[ \Delta x = 7 - 2 = 5 \]
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Calculate the difference in y-coordinates: \[ \Delta y = 3 - 5 = -2 \]
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Use the Pythagorean Theorem, which states that the distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ d = \sqrt{(\Delta x)^2 + (\Delta y)^2} \]
Substituting the values we computed: \[ d = \sqrt{(5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29} \]
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Calculate \(\sqrt{29}\): \[ d \approx 5.385 \]
Rounding to the nearest hundredth gives: \[ d \approx 5.39 \]
Therefore, the length between the points (2, 5) and (7, 3) is 5.39.
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