Question

Find the distance between (2, 3) and (−4, −9) . Round your answer to the nearest hundredth, if necessary.

Answer 1

To find the distance between the points \((2, 3)\) and \((-4, -9)\), we can use the distance formula:

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

Let \((x_1, y_1) = (2, 3)\) and \((x_2, y_2) = (-4, -9)\).

Now, we can substitute the coordinates into the formula:

\[
d = \sqrt{((-4) - (2))^2 + ((-9) - (3))^2}
\]

Calculating the differences:

\[
x_2 - x_1 = -4 - 2 = -6
\]
\[
y_2 - y_1 = -9 - 3 = -12
\]

Substituting these values back into the formula:

\[
d = \sqrt{(-6)^2 + (-12)^2}
\]

Calculating the squares:

\[
d = \sqrt{36 + 144}
\]

Now, add the squares:

\[
d = \sqrt{180}
\]

Now we simplify \(\sqrt{180}\):

\[
\sqrt{180} = \sqrt{36 \times 5} = 6\sqrt{5}
\]

Now, we can calculate the approximate numerical value of \(6\sqrt{5}\):

Using \(\sqrt{5} \approx 2.236\):

\[
d \approx 6 \times 2.236 \approx 13.416
\]

Rounding to the nearest hundredth gives:

\[
d \approx 13.42
\]

Thus, the distance between the points \((2, 3)\) and \((-4, -9)\) is approximately:

\[
\boxed{13.42}
\]