To find the distance between the points \((2, 5)\) and \((7, 3)\) using the distance formula, which is derived from the Pythagorean Theorem, we can use the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Where \((x_1, y_1)\) is \((2, 5)\) and \((x_2, y_2)\) is \((7, 3)\).
Now substitute the values into the formula:
\[ d = \sqrt{(7 - 2)^2 + (3 - 5)^2} \] \[ = \sqrt{(5)^2 + (-2)^2} \] \[ = \sqrt{25 + 4} \] \[ = \sqrt{29} \]
Calculating the square root of 29 gives us:
\[ d \approx 5.385 \]
Rounding to the nearest hundredth, the distance is approximately:
\[ d \approx 5.39 \]
Thus, the length between the points \((2, 5)\) and \((7, 3)\) is 5.39.
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