To find the distance between the points \((2, 5)\) and \((7, 3)\) using the Pythagorean Theorem, we first determine the horizontal and vertical distances:
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Calculate the differences in the coordinates:
- The horizontal distance (difference in x-coordinates): \[ x_2 - x_1 = 7 - 2 = 5 \]
- The vertical distance (difference in y-coordinates): \[ y_2 - y_1 = 3 - 5 = -2 \quad \text{(we will take the absolute value)} \]
- The absolute value of the vertical distance is: \[ |-2| = 2 \]
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Apply the Pythagorean Theorem: The distance \(d\) can be calculated using: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the distances: \[ d = \sqrt{(5)^2 + (2)^2} = \sqrt{25 + 4} = \sqrt{29} \]
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Calculate \(\sqrt{29}\): \[ \sqrt{29} \approx 5.385 \]
Rounding to the nearest hundredth gives us: \[ 5.39 \]
Thus, the length between the points \((2, 5)\) and \((7, 3)\) is approximately \(5.39\).
The correct response is 5.39.