To find the length between the two points \((7, 8)\) and \((3, 2)\) using the Pythagorean theorem, we can treat this as a right triangle where the legs are the differences in the x-coordinates and y-coordinates.
-
Calculate the difference in the x-coordinates: \[ \Delta x = 7 - 3 = 4 \]
-
Calculate the difference in the y-coordinates: \[ \Delta y = 8 - 2 = 6 \]
-
Now, apply the Pythagorean theorem: \[ d = \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{(4)^2 + (6)^2} \] \[ = \sqrt{16 + 36} = \sqrt{52} \]
-
Simplify the square root: \[ = \sqrt{52} = \sqrt{4 \times 13} = 2\sqrt{13} \]
-
Calculate the numerical value and round it to the nearest hundredth: \[ \sqrt{13} \approx 3.60555 \quad \text{(using a calculator)} \] \[ 2\sqrt{13} \approx 2 \times 3.60555 \approx 7.2111 \]
-
Round to the nearest hundredth: \[ \approx 7.21 \]
Thus, the length between the two points is approximately 7.21.