Final answer:
To find the distance between two points on a straight line, the Pythagorean theorem can be applied to calculate the hypotenuse of the right triangle formed by the horizontal and vertical distances. In graphical analysis, the slope of that line is found by determining 'rise over run' from two points on the line.
Step-by-step explanation:
To find the distance between two points connected by a line, you can employ the Pythagorean theorem. This theorem is very useful when the two points form a right triangle with the horizontal and vertical distances representing the two legs of the triangle. The formula a² + b² = c², where 'a' and 'b' are the lengths of the two legs and 'c' is the hypotenuse, lets us calculate the straight-line distance between the points.
For example, if you know the precise, straight-line distance on an athletics track between two points, you can measure the horizontal and vertical distances. If one leg of the right triangle is 3 meters and the other is 4 meters, according to the Pythagorean theorem, the distance is √(3² + 4²) = √(9 + 16) = √25 = 5 meters.
In another context, if you are looking to find the slope of a straight line on a graph between two points, you would calculate 'rise over run'. This involves picking two points on the line and finding the difference in their y-coordinates (rise) and x-coordinates (run). For instance, with the points (6, 10) and (2, 5), the slope would be (10 - 5) / (6 - 2) = 5 / 4.