Question

fill in the blank so that the resulting statement is true. the distance, d, between the points $(x_1,y_1)$ and $(x_2,y_2)$ in the rectangular coordinate system is d = the distance, d, between the points $(x_1,y_1)$ and $(x_2,y_2)$ in the rectangular coordinate system is $d=$

Explanation:

Step1: Recall the distance formula derivation

Consider a right - triangle formed by the two points \((x_1,y_1)\) and \((x_2,y_2)\). 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 \(d^{2}=(x_2 - x_1)^{2}+(y_2 - y_1)^{2}\).

Step3: Solve for \(d\)

Take the square root of both sides: \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\)

Answer:

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