To find the length between the points \((2, 7)\) and \((-2, 0)\) using the Pythagorean theorem, we first need to calculate the differences in the x-coordinates and the y-coordinates.
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Calculate the difference in the x-coordinates: \[ \Delta x = x_2 - x_1 = -2 - 2 = -4 \]
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Calculate the difference in the y-coordinates: \[ \Delta y = y_2 - y_1 = 0 - 7 = -7 \]
Now, we can use these differences to calculate the distance \(d\) between the two points using the distance formula, which is derived from the Pythagorean theorem:
\[ d = \sqrt{(\Delta x)^2 + (\Delta y)^2} \]
Substituting \(\Delta x\) and \(\Delta y\) into the formula:
\[ d = \sqrt{(-4)^2 + (-7)^2} \]
Calculating the squares:
\[ d = \sqrt{16 + 49} \]
Adding these values together:
\[ d = \sqrt{65} \]
Now, let's calculate \(\sqrt{65}\):
\[ \sqrt{65} \approx 8.0623 \]
Rounding to the nearest hundredth, the distance \(d\) is:
\[ \boxed{8.06} \]