Question

complete the sentence below. the distance d between two points ( p_1=(x_1,y_1) ) and ( p_2=(x_2,y_2) ) is ( d = ) the distance d between two points ( p_1=(x_1,y_1) ) and ( p_2=(x_2,y_2) ) is ( d = )

Explanation:

Step1: Recall the distance formula derivation

Consider a right - triangle with vertices \(P_1=(x_1,y_1)\), \(P_2=(x_2,y_2)\) and a point \((x_2,y_1)\). The horizontal side length is \(|x_2 - x_1|\) and the vertical side length is \(|y_2 - y_1|\).

Step2: Apply the Pythagorean theorem

By the Pythagorean theorem \(d^{2}=(x_2 - x_1)^{2}+(y_2 - y_1)^{2}\). Then take the square root of both sides.

Answer:

\(\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\)