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 y-coordinates.
Let \( (x_1, y_1) = (2, 7) \) and \( (x_2, y_2) = (-2, 0) \).
The differences are calculated as follows:
-
Difference in x-coordinates: \[ \Delta x = x_2 - x_1 = -2 - 2 = -4 \]
-
Difference in y-coordinates: \[ \Delta y = y_2 - y_1 = 0 - 7 = -7 \]
Next, we use the Pythagorean theorem to find the length \( d \) between the two points: \[ d = \sqrt{(\Delta x)^2 + (\Delta y)^2} \]
Now, substitute the differences we calculated: \[ d = \sqrt{(-4)^2 + (-7)^2} = \sqrt{16 + 49} = \sqrt{65} \]
Now, we can approximate \( \sqrt{65} \): \[ \sqrt{65} \approx 8.0623 \]
Rounding to the nearest hundredth: \[ d \approx 8.06 \]
Therefore, the length between the points (2, 7) and (−2, 0) is approximately 8.06.