Question

calculate the distance between the points n = (2, -5) and h = (7, -9) in the coordinate plane. give an exact answer (not a decimal approximation).

Explanation:

Step1: Recall the distance formula

The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Step2: Identify the coordinates

For points \( N = (2, -5) \) and \( H = (7, -9) \), we have \( x_1 = 2 \), \( y_1 = -5 \), \( x_2 = 7 \), \( y_2 = -9 \).

Step3: Substitute into the formula

First, calculate the differences: \( x_2 - x_1 = 7 - 2 = 5 \) and \( y_2 - y_1 = -9 - (-5) = -9 + 5 = -4 \).
Then, square these differences: \( (x_2 - x_1)^2 = 5^2 = 25 \) and \( (y_2 - y_1)^2 = (-4)^2 = 16 \).
Next, sum the squares: \( 25 + 16 = 41 \).
Finally, take the square root: \( d = \sqrt{41} \).

Answer:

\(\sqrt{41}\)