A Cartesian coordinate system in a plane consists of two perpendicular number lines, the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin (0, 0).
Any point P in the plane can be uniquely identified by an ordered pair of numbers (x, y), called its coordinates. The value 'x' is the perpendicular distance from the y-axis, and 'y' is the perpendicular distance from the x-axis.
The distance d between two points P(x₁, y₁) and Q(x₂, y₂) is found using the Pythagorean theorem.
Distance Formula: d = sqrt( (x₂ - x₁)² + (y₂ - y₁)² )
Find the distance between A(2, 3) and B(5, -1).
Solution:
d = sqrt( (5 - 2)² + (-1 - 3)² )
d = sqrt( (3)² + (-4)² )
d = sqrt( 9 + 16 ) = sqrt(25) = 5 units.
This formula finds the coordinates of a point R(x, y) that divides the line segment joining P(x₁, y₁) and Q(x₂, y₂) in a specific ratio, m : n.
The point R lies *between* P and Q.
R(x, y) = ( (mx₂ + nx₁) / (m + n) , (my₂ + ny₁) / (m + n) )
The point R lies *outside* the line segment PQ (on the line extended).
R(x, y) = ( (mx₂ - nx₁) / (m - n) , (my₂ - ny₁) / (m - n) )
This is a special case of internal division where the ratio is 1 : 1 (so m=1, n=1).
Mid-Point = ( (x₁ + x₂) / 2 , (y₁ + y₂) / 2 )
If the vertices of a triangle are A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃), its area is given by the absolute value of:
Area = (1/2) | x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) |
This can be more easily remembered using the determinant "Shoelace Formula":
Area = (1/2) | (x₁y₂ + x₂y₃ + x₃y₁) - (y₁x₂ + y₂x₃ + y₃x₁) |
Condition for Collinearity: Three points are collinear (lie on the same straight line) if the area of the triangle they form is zero.
To find the area of a quadrilateral with vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃), and D(x₄, y₄) in order:
Note: Be sure to take the vertices in order (clockwise or counter-clockwise) when applying the shoelace formula directly to the quadrilateral.
This is an alternative system for locating points in a plane. A point is defined by (r, θ) where:
r = The radius vector, or the directed distance from the origin (called the pole) to the point.θ = The vectorial angle, or the angle measured counter-clockwise from the initial line (which is the positive x-axis) to the radius vector.Given a point (x, y), we can find (r, θ) using the relationships from the Pythagorean theorem.
r = sqrt(x² + y²)
θ = tan⁻¹(y / x)
tan⁻¹ function on a calculator will only give answers in Quadrant I or IV. You may need to add or subtract 180° (or π) based on the signs of x and y. For example, (-1, -1) is in Q3, but tan⁻¹(-1/-1) = tan⁻¹(1) = 45°, which is in Q1. The correct angle is 45° + 180° = 225°.
Given a point (r, θ), we can find (x, y) using basic trigonometry.
x = r cos(θ)
y = r sin(θ)
Convert the polar point (4, 30°) to Cartesian.
Solution:
x = 4 cos(30°) = 4 (sqrt(3) / 2) = 2*sqrt(3)
y = 4 sin(30°) = 4 (1 / 2) = 2
The Cartesian point is (2*sqrt(3), 2).
d = sqrt(Δx² + Δy²).(mx₂ + nx₁) / (m + n). For external, just change all '+' to '-'.(x₁ + x₂) / 2.x = r cos(θ)y = r sin(θ)r² = x² + y²