Coordinate Geometry-Vocabulary List

Coordinate Geometry Vocabulary

  • Cartesian Plane: A plane defined by two perpendicular number lines, called the x-axis and y-axis, intersecting at a point called the origin.
  • Origin: The point where the x-axis and y-axis intersect, represented as (0,0).
  • Quadrants: The four regions of the Cartesian plane, divided by the x-axis and y-axis. They are labeled as Quadrant I, II, III, and IV.
  • Coordinates: An ordered pair of numbers (x,y) used to locate a point on the Cartesian plane, where x is the horizontal position (x-coordinate) and y is the vertical position (y-coordinate).
  • Distance Formula: A formula used to find the distance between two points (x1,y1) and (x2,y2) on the Cartesian plane:

    d=(x2x1)2+(y2y1)2

  • Midpoint Formula: A formula used to find the midpoint of a line segment connecting two points (x1,y1) and (x2,y2):

    (x1+x22,y1+y22)

  • Slope: A measure of the steepness or incline of a line, defined as the ratio of the vertical change to the horizontal change between two points on the line:

    m=y2y1x2x1

  • Slope-Intercept Form: The equation of a line written as y=mx+c, where m is the slope and c is the y-intercept (the point where the line crosses the y-axis).
  • Point-Slope Form: The equation of a line that passes through a point (x1,y1) with slope m:

    yy1=m(xx1)

  • Y-Intercept: The point where a line crosses the y-axis, represented as (0,y).
  • X-Intercept: The point where a line crosses the x-axis, represented as (x,0).
  • Equation of a Circle: The equation of a circle with center (a,b) and radius r is:

    (xa)2+(yb)2=r2

  • Collinear Points: Points that lie on the same straight line.
  • Parallel Lines: Lines in the same plane that never intersect, having the same slope but different y-intercepts.
  • Perpendicular Lines: Lines that intersect at a right angle (90°). The slopes of perpendicular lines are negative reciprocals of each other:

    m1×m2=1

  • Isosceles Triangle: A triangle with two equal sides. In coordinate geometry, the distance formula can be used to check whether two sides of a triangle are equal.
  • Equilateral Triangle: A triangle where all three sides are of equal length. Again, the distance formula is used to verify this.
  • Right-Angled Triangle: A triangle with one angle equal to 90°. The Pythagorean theorem is often used to verify right-angled triangles:

    c2=a2+b2

  • Collinearity Test: A method of determining if three points are collinear by calculating the slopes of the line segments between them. If the slopes are equal, the points are collinear.
  • Distance between Parallel Lines: The shortest distance between two parallel lines is given by:

    d=|c1c2|a2+b2

    Where c1 and c2 are the constants in the equations of the lines, and a and b are coefficients of x and y in the general line equation.
  • Centroid: The point where the medians of a triangle intersect, calculated as the average of the x-coordinates and y-coordinates of the triangle's vertices:

    G=(x1+x2+x33,y1+y2+y33)

  • Pythagorean Theorem: A fundamental relation in a right-angled triangle between the lengths of the hypotenuse and the other two sides:

    c2=a2+b2

  • Inclination of a Line: The angle that a line makes with the positive x-axis, usually denoted by θ. The slope m of the line is related to the inclination angle by:

    m=tan(θ)

  • General Form of a Line: The equation of a line in the form:

    ax+by+c=0

    where a, b, and c are constants.
  • Vector Equation of a Line: The vector form of the equation of a line can be written as:

    r=r0+td

    where r0 is a point on the line, d is the direction vector, and t is a scalar.
  • Area of a Triangle (using coordinates): The area of a triangle with vertices (x1,y1), (x2,y2), and (x3,y3) can be calculated using the formula:

    Area=12|x1(y2y3)+x2(y3y1)+x3(y1y2)|

  • Centroid of a Triangle: The point where the three medians of a triangle intersect. The centroid divides each median into a 2:1 ratio, with the longer part closer to the vertex.
  • Circumcenter: The point where the perpendicular bisectors of the sides of a triangle intersect. It is the center of the circumcircle, the circle that passes through all three vertices of the triangle.
  • Orthocenter: The point where the altitudes of a triangle intersect. The altitude is the perpendicular segment from a vertex to the opposite side.