In coordinate geometry, the distance formula is used to find the distance between two points on a coordinate plane. Given two points with coordinates (x₁, y₁) and (x₂, y₂), the distance formula is derived from the Pythagorean theorem:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Let's consider an example: given points A(1, 2) and B(4, 6), we can find the distance between these points using the distance formula. Plugging in the values, we get:

d = √[(4 - 1)² + (6 - 2)²] = √[3² + 4²] = √[9 + 16] = √25 = 5

The distance between points A and B is 5 units.

Moving on to the midpoint formula, it is used to find the coordinates of the midpoint between two points. Given points A(x₁, y₁) and B(x₂, y₂), the midpoint formula is:

M((x₁ + x₂)/2, (y₁ + y₂)/2)

To find the midpoint between A(1, 2) and B(4, 6), we substitute the values into the formula:

M((1 + 4)/2, (2 + 6)/2) = M(5/2, 8/2) = M(2.5, 4)

The coordinates of the midpoint are (2.5, 4).

Both the distance formula and the midpoint formula are valuable tools in coordinate geometry, allowing us to calculate distances and find midpoints between points on a coordinate plane.