Apply the Pythagorean Theorem to find the length between the two points (2, -3) and (5, 5). Round to the nearest hundredth, if necessary.

A. 7.62
B. 11
C. 73
D. 8.54

1 answer

To find the length between the two points \((2, -3)\) and \((5, 5)\), we can use the distance formula, which is derived from the Pythagorean theorem. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Here, our points are \((x_1, y_1) = (2, -3)\) and \((x_2, y_2) = (5, 5)\).

First, we calculate the differences in the coordinates:

\[ x_2 - x_1 = 5 - 2 = 3 \] \[ y_2 - y_1 = 5 - (-3) = 5 + 3 = 8 \]

Now we input these differences into the distance formula:

\[ d = \sqrt{(3)^2 + (8)^2} \] \[ d = \sqrt{9 + 64} \] \[ d = \sqrt{73} \]

Next, we need to calculate \(\sqrt{73}\) and round it to the nearest hundredth:

\[ \sqrt{73} \approx 8.544 \]

Rounding this to the nearest hundredth gives us:

\[ d \approx 8.54 \]

So the answer is \( \boxed{8.54} \).