Question

calculate the distance between the points ( n = (3, -4) ) and ( p = (7, -9) ) in the coordinate plane. give an exact answer (not a decimal approximation).

Explanation:

Step1: Recall distance formula

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Here, \(x_1 = 3\), \(y_1 = -4\), \(x_2 = 7\), \(y_2 = -9\).

Step2: Substitute values into formula

First, calculate \(x_2 - x_1 = 7 - 3 = 4\) and \(y_2 - y_1 = -9 - (-4) = -9 + 4 = -5\). Then, substitute into the formula: \(d = \sqrt{(4)^2 + (-5)^2}\).

Step3: Simplify the expression

Calculate the squares: \(4^2 = 16\) and \((-5)^2 = 25\). Then add them: \(16 + 25 = 41\). So, \(d = \sqrt{41}\).

Answer:

\(\sqrt{41}\)