Question

Apply the Pythagorean Theorem to find the length between (2,7) and (−2, 0). Round your answer to the nearest hundredth, if necessary

Answer 1

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:

1. Difference in x-coordinates:
\[
\Delta x = x_2 - x_1 = -2 - 2 = -4
\]

2. 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**.