To determine whether the statement "To find the distance between any two points in the plane, you can always use the distance formula" is true or false, let's explore what the distance formula is and its applicability.
The distance formula is derived from the Pythagorean Theorem and is used to calculate the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in a Cartesian plane. The formula is given by:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
This formula works due to the following reasoning:
1. The difference [tex]\((x_2 - x_1)\)[/tex] gives the horizontal distance between the two points.
2. The difference [tex]\((y_2 - y_1)\)[/tex] gives the vertical distance between the two points.
3. By applying the Pythagorean Theorem, these horizontal and vertical distances form the legs of a right triangle, where the distance we are seeking is the hypotenuse.
Given this, we see that the distance formula is a universal method to find the distance between any two points in a plane as long as their coordinates are known. It provides a precise and straightforward way to calculate the Euclidean distance.
Therefore, the statement "To find the distance between any two points in the plane, you can always use the distance formula" is:
A. True
