To find the distance between the points \((2, 3)\) and \((-4, -9)\), we can use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \((x_1, y_1) = (2, 3)\) and \((x_2, y_2) = (-4, -9)\).
Now, we can plug in the coordinates into the formula:
\[ d = \sqrt{((-4) - 2)^2 + ((-9) - 3)^2} \]
Calculating each part:
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Calculate \(x_2 - x_1\): \[ -4 - 2 = -6 \] So, \((-6)^2 = 36\).
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Calculate \(y_2 - y_1\): \[ -9 - 3 = -12 \] So, \((-12)^2 = 144\).
Now plug these into the distance formula:
\[ d = \sqrt{36 + 144} = \sqrt{180} \]
Now, simplify \(\sqrt{180}\):
\[ \sqrt{180} = \sqrt{36 \times 5} = 6\sqrt{5} \]
Using a calculator to get the numerical value:
\[ 6\sqrt{5} \approx 6 \times 2.236 \approx 13.416 \]
Rounding to the nearest hundredth, we get:
\[ d \approx 13.42 \]
Thus, the distance between the points \((2, 3)\) and \((-4, -9)\) is approximately 13.42.