Question

which of the following statements is not true?
choose the incorrect statement below
a. for any real numbers x₁ and x₂, the expression |x₂ - x₁|² is equivalent to the expression (x₂ - x₁)²
b. the distance between two points is always greater than or equal to zero
c. it is possible for the distance between two points to be negative
d. the pythagorean theorem is used to derive the distance formula

Explanation:

Step1: Analyze option A

For any real - numbers \(x_1\) and \(x_2\), \(\vert x_2 - x_1\vert^2=(x_2 - x_1)^2\) since \(\vert a\vert^2=a^2\) for any real number \(a=x_2 - x_1\).

Step2: Analyze option B

By the definition of distance, the distance between two points (in any metric space, especially in the real - number line or in the coordinate plane) is always non - negative.

Step3: Analyze option C

The distance between two points represents a length, and length cannot be negative. So, it is not possible for the distance between two points to be negative.

Step4: Analyze option D

The distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) in the coordinate plane is derived from the Pythagorean theorem.

Answer:

C. It is possible for the distance between two points to be negative