Coordinate Geometry - Distance Formula

The Cartesian Plane: A two-dimensional coordinate system formed by the intersection of a horizontal line ($x$-axis) and a vertical line ($y$-axis). Visually, these axes meet at a point called the Origin $(0, 0)$ and divide the plane into four quadrants.

Coordinates of a Point: Every point is represented by an ordered pair $(x, y)$, where $x$ is the abscissa (distance from the $y$-axis) and $y$ is the ordinate (distance from the $x$-axis). Visually, to locate $(3, 4)$, you move 3 units right and 4 units up from the origin.

Geometric Interpretation of Distance: The distance between two points $P$ and $Q$ is the length of the straight line segment joining them. If you draw horizontal and vertical lines through $P$ and $Q$, they form a right-angled triangle where the segment $PQ$ is the hypotenuse.

The Distance Formula: Derived from the Pythagorean theorem, the distance $d$ between points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated as the square root of the sum of the squares of the differences in their $x$ and $y$ coordinates.

Distance from the Origin: A specialized case of the distance formula where one point is $(0, 0)$. For any point $P(x, y)$, the distance from the origin is simply the square root of the sum of the squares of its coordinates.

Collinearity of Points: Three points $A$, $B$, and $C$ are collinear (lying on the same straight line) if the sum of the lengths of any two segments equals the length of the third segment, such as $AB + BC = AC$.

Classification of Triangles: By calculating the distances between vertices, we can identify triangle types. For instance, an equilateral triangle has three equal side lengths, while an isosceles triangle has two.

Properties of Quadrilaterals: The distance formula helps verify shapes. A square is confirmed if all four sides are equal and both diagonals are equal in length. In a rectangle, opposite sides are equal and diagonals are equal.