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The study of geometric figures by mapping them on coordinate axes is known as coordinate geometry. Straight lines, circles, curves, hyperbolas, ellipses, and polygons may all be drawn readily and scaled in the coordinate axes. Further coordinate geometry aids in algebraic work and the study of the properties of geometric figures using the coordinate system. Coordinate geometry is a field of mathematics that assists in displaying geometric forms on a two-dimensional plane and learning the attributes of these figures. In this section, we will try to learn about the coordinate plane and the coordinates of a point in order to get a basic grasp of coordinate geometry.
Coordinate Plane: A cartesian plane divides planar space into two dimensions, making it easier to identify points. It is also termed a coordinate plane. The coordinate plane’s axes are the horizontal x-axis and a vertical y-axis. These coordinate axes divide the plane into four quadrants, and the intersection of these axes is known as the origin (0, 0). Furthermore, each point in the coordinate plane is denoted by a point (x, y), where the x value represents the point’s location with respect to the x-axis and the y-value shows the location of the point in relation to the y-axis. A coordinate is an address that aids in the location of a point in space. The coordinates of a point in a 2D space are (x, y). Let us consider these two crucial phrases.
- Abscissa: It is the x value in the location (x, y), and also the distance along the x-axis from the origin.
- Ordinate: It is the y-value in the point (x, y), as well as the point’s perpendicular distance from the x-axis, which is parallel to the y-axis.
The coordinates of a point may be used to conduct a variety of operations such as determining distance, midpoint, the slope of a line, and equation of a line.
How to solve for geometry:
Coordinate geometry formulae make it easier to prove the many qualities of lines and figures represented by coordinate axes. Coordinate geometry formulae include the distance formula, slope formula, midpoint formula, section formula, and line equation. In the next paragraph let’s discuss some geometry formulas.
Distance Formula: Distance between two coordinates (X1, Y1) and (X2, Y2) is equal to the square root of the sum of the squares of the difference between their x and y coordinates. The following is the formula for calculating the distance between two places:
D = √(X2-X1)2+(Y2-Y1)2
Slope Formula: The inclination of a line is represented by its slope. The slope can be computed by picking any two points on the line or by calculating the angle formed by the line with the positive x-axis. m = Tan θ is the slope of a line inclined at an angle with the positive x-axis. The slope of a line connecting two points (X1, Y1) and (X2, Y2) is equal to m = (Y2 – Y1)/(X2 – X1 ).
m = Tan θ
m = (Y2 – Y1)/(X2 – X1 ).
Mid-Point Formula: The formula for finding the midpoint of the line connecting the points (X1, Y1) and (X2, Y2) creates a new point with an abscissa equal to the average of the x values of the two supplied points and an ordinate equal to the average of the y values of the two given points. The midway is placed exactly between the two locations and lies on the line connecting them.
(X,Y) = ((X1 +X2)/2, (Y1+Y2)/2)
Section Formula in Coordinate Geometry: The section formula may be used to get the coordinates of a point that splits the line segment connecting the points (X1, Y1) and (X2, Y2) in the ratio m:n. The point separating the supplied two points is located on the line connecting the two points and can be found either between the two points or outside the line segment between the points.
(X,Y) = [(mX2 + nX1)/(m+n), (mY2 + nY1)/(m+n)]
The centroid of Triangle: The centroid of a triangle is a point at which the triangle’s medians connect. A median is a line that links the vertex of a triangle to the midway of the opposing side. The following formula yields the centroid of a triangle with vertices A (X1 + Y2), B(X2 + Y2), and C(X3 + Y3).
(X,Y) = [( X1 + X2 +X3 )/3, (Y1 + Y2 + Y3)/3]
How to find the equation of a line in coordinate geometry: With the use of a basic linear equation, this line equation depicts all of the points on the line. The conventional form of a line equation is axe + by + c= 0. There are several techniques for determining a line’s equation. The slope-intercept form of the equation of a line (y = mx + c) is another essential form of the equation of a line. Here, m denotes the line’s slope, while c denotes the line’s y-intercept.
y = mx + c
Area of a triangle using coordinate geometry formula:
The area of a triangle with vertices A(X1, Y1), B(X2, Y2), and C(X3, Y3) is calculated using the formula below. This method for calculating the area of a triangle is applicable to all forms of triangles.
Area of a Triangle = 1/2[ X1 (Y2 – Y3) + X2 (Y3 – Y1) + X3 (Y1 – Y2)]
