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:
\[ \boxed{13.42} \]