To find the distance between the points \((0, 6)\) and \((-3, -2)\) using the Pythagorean Theorem, we can treat the distance as the hypotenuse of a right triangle.
First, we calculate the differences in the x-coordinates and y-coordinates:
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The difference in the x-coordinates is: \[ x_2 - x_1 = -3 - 0 = -3 \]
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The difference in the y-coordinates is: \[ y_2 - y_1 = -2 - 6 = -8 \]
Now, we take the absolute values of these differences: \[ |x_2 - x_1| = 3 \] \[ |y_2 - y_1| = 8 \]
We can now use the Pythagorean Theorem to find the distance \(d\) between the points: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the values we found: \[ d = \sqrt{(-3)^2 + (-8)^2} \] \[ d = \sqrt{9 + 64} \] \[ d = \sqrt{73} \]
Now, we compute \(\sqrt{73}\): \[ d \approx 8.544 \] Rounding to the nearest hundredth gives: \[ d \approx 8.54 \]
Therefore, the length between the points \((0, 6)\) and \((-3, -2)\) is approximately 8.54.