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, which is given by:

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

Let's assign the points:

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

Now we can substitute these values into the formula.

1. Calculate \( x_2 - x_1 \) and \( y_2 - y_1 \):

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

2. Substitute into the distance formula:

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

3. Calculate the squares:

\[
(-6)^2 = 36
\]
\[
(-12)^2 = 144
\]

4. Add the squares:

\[
d = \sqrt{36 + 144} = \sqrt{180}
\]

5. Simplify \( \sqrt{180} \):

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

6. Use a calculator to find \( \sqrt{5} \):

\[
\sqrt{5} \approx 2.236
\]

7. Now compute \( 6\sqrt{5} \):

\[
6 \times 2.236 \approx 13.416
\]

Finally, rounding to the nearest hundredth:

\[
d \approx 13.42
\]

Thus, the distance between the points \( (2, 3) \) and \( (-4, -9) \) is approximately \( \boxed{13.42} \).