THE distance between two points can be obtained by calculating or measuring the length of the straight segment that connects them. To calculate your length, we can make use of the Geometry THEanalytic and find a formula capable of determining the distancein betweentwopoints using your coordinates. Considering points A (xTHEyTHE) and B(xByB), this formula takes the following form:
Demonstration
To demonstrate this formula, we need to mark the points A(xTHEyTHE) and B(xByB) at the Cartesian plane. Once that's done, build the straight segment that connects them. remember that the distancein betweentwopoints is the shortest distance between them.
In the figure above, we mark points A and B and highlight their coordinates on the x and y axes. Note that to mark these coordinates, we ended up building the triangle ABC, which is rectangle and whose hypotenuse is the AB segment. That way we can use the Pythagorean theorem to find the length of that segment.
Before that, however, it is necessary to know the length of the
segments AC and BC, which are the legs of this triangle. Segment AC measures xB – xTHE, while the BC segment measures yB - yTHE. At the Pythagorean theorem, we will have:
like the length of the segment AB is just the distance between points A and B, we conclude that:
Examples:
1ºExample – Determine the distance between points A(2, 4) and B(3, 8).
Solution: Using the formula of distancein betweentwopoints, we will have:
2nd Example – Determine the length of the straight segment of the following image:
Solution: Note that the coordinates of the points are: A(1, 1) and B(5, 4). So, just use the formula to determine the distance between points A and B, which is just the length of the segment AB.
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-distancia-entre-dois-pontos.htm