To find the distance between the points \((2, 5)\) and \((7, 3)\) using the Pythagorean Theorem, we can follow these steps:
-
First, determine the differences in the \(x\) and \(y\) coordinates: \[ \Delta x = x_2 - x_1 = 7 - 2 = 5 \] \[ \Delta y = y_2 - y_1 = 3 - 5 = -2 \]
-
Now, use the distance formula, which is derived from the Pythagorean Theorem: \[ d = \sqrt{(\Delta x)^2 + (\Delta y)^2} \] Plugging in the values we found: \[ d = \sqrt{(5)^2 + (-2)^2} \]
-
Calculate the squares: \[ d = \sqrt{25 + 4} = \sqrt{29} \]
-
Finally, calculate \(\sqrt{29}\) and round to the nearest hundredth: \[ \sqrt{29} \approx 5.385 \]
Rounded to the nearest hundredth, the length between the points \((2, 5)\) and \((7, 3)\) is approximately: \[ \boxed{5.39} \]