The distance formula allows us to find the distance between two points, A and B, on a coordinate plane. It is derived from the Pythagorean theorem and is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
To understand this formula, let's consider an example. Suppose we have two points, A(2, 3) and B(5, 7). We can plug these values into the distance formula as follows:
d = sqrt((5 - 2)^2 + (7 - 3)^2) = sqrt(9 + 16) = sqrt(25) = 5
Thus, the distance between points A and B is 5 units.
The midpoint formula allows us to find the coordinates of the midpoint, M, of a line segment with endpoints A and B. It is given by:
M = ((x1 + x2) / 2, (y1 + y2) / 2)
Using the previous example, we can find the midpoint of the line segment AB as:
M = ((2 + 5) / 2, (3 + 7) / 2) = (3.5, 5)
By applying these formulas, we can easily calculate distances and find midpoints in coordinate geometry problems.