Questions: Find the distance between the points. Give an exact answer and an approximation to three decimal places. (2,-9) and (-3,-19)

Solution Steps

To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a 2D plane, we use the distance formula:

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

This formula calculates the Euclidean distance by finding the difference in the x-coordinates and y-coordinates, squaring these differences, summing them, and then taking the square root of the result.

The given points are \((2, -9)\) and \((-3, -19)\).

To find the distance \(d\) between the points, use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substitute the coordinates into the formula: \[ d = \sqrt{((-3) - 2)^2 + ((-19) - (-9))^2} \]

Calculate the differences in the coordinates: \[ x_2 - x_1 = -3 - 2 = -5 \] \[ y_2 - y_1 = -19 + 9 = -10 \]

Square each of the differences: \[ (-5)^2 = 25 \] \[ (-10)^2 = 100 \]

Add the squared differences: \[ 25 + 100 = 125 \]

Take the square root of the sum to find the distance: \[ d = \sqrt{125} \approx 11.1803 \]

Final Answer