Coordinate Geometry explores the relationship between algebra and geometry through the use of coordinate systems to represent geometric shapes and solve problems.
Coordinate Geometry – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class X in Mathematics.
This one-pager compiles key formulas and equations from the Coordinate Geometry chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
Distance between two points P(x₁, y₁) and Q(x₂, y₂): √((x₂ - x₁)² + (y₂ - y₁)²)
Calculates the straight-line distance between two points in a plane. x₁, y₁ are coordinates of point P; x₂, y₂ are coordinates of point Q. Essential for finding lengths in geometry problems.
Mid-point formula: ((x₁ + x₂)/2, (y₁ + y₂)/2)
Finds the midpoint of a line segment joining two points P(x₁, y₁) and Q(x₂, y₂). Useful in dividing a line segment into two equal parts.
Section formula (internal division): ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂))
Divides a line segment internally in the ratio m₁:m₂. m₁ and m₂ are the ratios; (x₁, y₁) and (x₂, y₂) are the endpoints. Key for problems involving ratio division.
Area of a triangle using coordinates: ½ |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
Calculates the area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃). Absolute value ensures the area is non-negative.
Slope of a line (m): (y₂ - y₁)/(x₂ - x₁)
Measures the steepness of a line joining two points (x₁, y₁) and (x₂, y₂). Positive slope indicates an upward trend; negative, downward.
Equation of a line: y - y₁ = m(x - x₁)
Finds the equation of a line with slope m passing through point (x₁, y₁). Fundamental for graphing linear equations.
Distance of a point P(x, y) from origin: √(x² + y²)
Special case of the distance formula where one point is the origin (0,0). Useful in problems involving central symmetry.
Centroid of a triangle: ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)
The intersection point of the medians of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃). Represents the average position of the three vertices.
Collinearity condition: |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)| = 0
Determines if three points are collinear (lie on the same straight line). Derived from the area of a triangle formula.
Section formula (external division): ((m₁x₂ - m₂x₁)/(m₁ - m₂), (m₁y₂ - m₂y₁)/(m₁ - m₂))
Divides a line segment externally in the ratio m₁:m₂. Useful for problems where the dividing point lies outside the segment.
Equations
Distance from point (x, y) to line Ax + By + C = 0: |Ax + By + C|/√(A² + B²)
Calculates the perpendicular distance from a point to a line. A, B, C are coefficients of the line equation; (x, y) is the point. Essential for shortest distance problems.
Equation of x-axis: y = 0
The x-axis is the horizontal axis where all points have a y-coordinate of 0. Fundamental for coordinate system basics.
Equation of y-axis: x = 0
The y-axis is the vertical axis where all points have an x-coordinate of 0. Fundamental for coordinate system basics.
Parallel lines condition: m₁ = m₂
Two lines are parallel if their slopes m₁ and m₂ are equal. Useful in identifying parallel lines without graphing.
Perpendicular lines condition: m₁ * m₂ = -1
Two lines are perpendicular if the product of their slopes m₁ and m₂ is -1. Key for problems involving right angles.
General equation of a line: Ax + By + C = 0
Represents any straight line in the plane. A, B, C are constants. Standard form for linear equations.
Slope-intercept form: y = mx + c
m is the slope; c is the y-intercept. Simplifies graphing linear equations by directly showing slope and y-intercept.
Intercept form of a line: x/a + y/b = 1
a and b are x and y intercepts respectively. Useful for quickly sketching lines when intercepts are known.
Point-slope form: y - y₁ = m(x - x₁)
Derived from the slope definition. Directly relates a point (x₁, y₁) and slope m to the line's equation.
Two-point form of a line: (y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁)
Uses two points (x₁, y₁) and (x₂, y₂) to define a line. Alternative to point-slope form when two points are known.
Explore the world of Polynomials, understanding their types, degrees, and operations to solve algebraic expressions and equations effectively.
Explore the methods to solve a pair of linear equations in two variables, including graphical, substitution, elimination, and cross-multiplication techniques.
Explore the world of quadratic equations, learning to solve them using various methods like factoring, completing the square, and the quadratic formula.
A chapter that explores sequences where each term after the first is obtained by adding a constant difference, focusing on their properties, nth term, and sum formulas.
Explore the properties, types, and theorems related to triangles, including congruence and similarity, to solve geometric problems effectively.
Explore the basics of trigonometry, including angles, triangles, and the fundamental trigonometric ratios: sine, cosine, and tangent.
Explore real-world applications of trigonometry in measuring heights, distances, and angles in various fields such as astronomy, navigation, and architecture.
Explore the properties, theorems, and applications of circles in geometry, including tangents, chords, and angles subtended by arcs.
Explore the concepts of calculating areas related to circles, including sectors, segments, and combinations with other geometric shapes.
Explore the concepts of calculating surface areas and volumes of various geometric shapes, including cubes, cylinders, cones, and spheres, to solve real-world problems.
