Flash Cards: Coordinate Geometry

Coordinates are numeric values that determine the position of a point in a plane using an ordered pair (x, y).

The abscissa is the x-coordinate, representing the distance of the point from the y-axis.

If points are A(a, 0) and B(b, 0), distance AB = |b - a|.

If points are C(0, c) and D(0, d), distance CD = |d - c|.

Use the Pythagorean theorem to calculate the distance from each point's projections on the axes.

Distance PQ = √((6 - 4)² + (8 - 6)²) = √8 = 2√2.

Points are collinear if they lie on a straight line, verified by distance relationships.

Calculate distances; if the sum of the distances between them equals the third side, they are collinear.

The midpoint M between two points P(x₁, y₁) and Q(x₂, y₂) is: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).

It calculates the coordinates of a point dividing a segment in a given ratio m:n as: ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

The distance OP is given by OP = √(x² + y²).

Points on the x-axis have coordinates of the form (x, 0).

Points on the y-axis have coordinates of the form (0, y).

Calculate distances; if the square of the longest side equals the sum of squares of the others, it's a right triangle.

The standard form is Ax + By + C = 0, where A, B cannot both be zero.

It connects algebra to geometry, enabling the study of geometric figures using algebraic equations.

It is applied in physics, engineering, navigation, and art for modeling and analyzing shapes.