Question

Find the distance between the points (9, -6) and (3,0).
Write your answer as a whole number or a fully simplified radical expression. Do not round.

Answer 1

To find the distance between the points \((9, -6)\) and \((3, 0)\), we can use the distance formula, which is given by:

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

Here, \((x_1, y_1) = (9, -6)\) and \((x_2, y_2) = (3, 0)\).

Now, plug in the values:

\[
d = \sqrt{(3 - 9)^2 + (0 - (-6))^2}
\]

Calculating each part:

1. Calculate \(x_2 - x_1\):

\[
3 - 9 = -6
\]

2. Calculate \(y_2 - y_1\):

\[
0 - (-6) = 0 + 6 = 6
\]

Now substitute these back into the distance formula:

\[
d = \sqrt{(-6)^2 + (6)^2}
\]

Calculating the squares:

\[
(-6)^2 = 36
\]
\[
(6)^2 = 36
\]

Now we can add these results:

\[
d = \sqrt{36 + 36} = \sqrt{72}
\]

Next, simplify \(\sqrt{72}\):

\[
\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}
\]

Thus, the distance between the points \((9, -6)\) and \((3, 0)\) is:

\[
\boxed{6\sqrt{2}}
\]