Question

question 4 of 10
the distance formula is derived from the pythagorean theorem.

a. true

b. false

Explanation:

To derive the distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) in a coordinate plane, we can form a right triangle where the horizontal leg has length \(|x_2 - x_1|\), the vertical leg has length \(|y_2 - y_1|\), and the distance between the points is the hypotenuse. By the Pythagorean theorem (\(a^{2}+b^{2}=c^{2}\), where \(a\) and \(b\) are the legs and \(c\) is the hypotenuse), we substitute \(a = |x_2 - x_1|\), \(b=|y_2 - y_1|\) and \(c = d\) (distance). Then \(d^{2}=(x_2 - x_1)^{2}+(y_2 - y_1)^{2}\), and taking the square root gives the distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\). So the distance formula is derived from the Pythagorean theorem.

Answer:

A. True