Unit 1: Coordinate Systems and Basic Formulae
Table of Contents
Coordinates
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.
Distance Between Two Points
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 )
Example:
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.
Section Formula
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.
Internal Division
The point R lies *between* P and Q.
R(x, y) = ( (mx₂ + nx₁) / (m + n) , (my₂ + ny₁) / (m + n) )
External Division
The point R lies *outside* the line segment PQ (on the line extended).
R(x, y) = ( (mx₂ - nx₁) / (m - n) , (my₂ - ny₁) / (m - n) )
Mid-Point Formula
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 )
Area of Triangle and Quadrilateral
Area of a Triangle (Determinant Form)
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.
Area of a Quadrilateral
To find the area of a quadrilateral with vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃), and D(x₄, y₄) in order:
- Draw a diagonal, for example, AC.
- This divides the quadrilateral into two triangles: ΔABC and ΔADC.
- Calculate the area of ΔABC and the area of ΔADC using the formula above.
- Add the two areas together.
Note: Be sure to take the vertices in order (clockwise or counter-clockwise) when applying the shoelace formula directly to the quadrilateral.
Polar Coordinates
Definition
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.
Conversion: Cartesian to Polar
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°. Conversion: Polar to Cartesian
Given a point (r, θ), we can find (x, y) using basic trigonometry.
x = r cos(θ)
y = r sin(θ)
Example:
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).
Unit 1: Exam Quick Tips
- Distance Formula:
d = sqrt(Δx² + Δy²). - Section Formula:
(mx₂ + nx₁) / (m + n). For external, just change all '+' to '-'. - Mid-point: Average of x's, average of y's.
(x₁ + x₂) / 2. - Area of Triangle: Use the shoelace/determinant method. It's faster and less prone to sign errors.
- Collinearity: Just show the Area = 0.
- Polar Conversion:
x = r cos(θ)y = r sin(θ)r² = x² + y²