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Introduction to coordinate geometry
Every location on the planet has coordinates that allow us to quickly locate it on a world map. Our planet’s coordinate system is made up of imaginary lines known as latitudes and longitudes. The starting lines of this coordinate system are zero degrees ‘Greenwich Longitude’ and zero degrees ‘Equator Latitude.’ The coordinate axes, with the horizontal x-axis and the vertical y-axis, are used to locate a point in a plane or on a sheet of paper.
Cartesian plane
The study of geometric figures by using coordinate axes is known as coordinate geometry. Straight lines, curves, circles, ellipses, hyperbolas, and polygons can all be readily drawn and scaled in the coordinate axes. Furthermore, using the coordinate system to work algebraically and explore the attributes of geometric figures is known as coordinate geometry.
Plane of Coordination
A Cartesian plane divides plane space into two dimensions, making it easier to locate points. The coordinate plane is another name for it. The horizontal x-axis and the vertical y-axis are the two axes of the coordinate plane. The origin is the place where these coordinate axes connect, dividing the plane into four quadrants (0, 0). Furthermore, any point in the coordinate plane is represented by a point (x, y), where the x value represents the point’s position relative to the x-axis and the y value represents the point’s position relative to the y-axis.
Coordinate geometry formulae and Theorems
Distance Formula in Coordinate Geometry
The square root of the sum of the squares of the difference between the x and y coordinates of the two provided points is the distance between two points ( x2, y2) and x1, y1 ). The following are for calculating the distance between two points.
Slope formula
The inclination of a line is measured by its slope. The slope can be estimated by picking any two locations on the line and measuring the angle formed by the line with the positive x-axis. m = Tan theta is the slope of a line that is inclined at an angle to the positive x-axis. The slope of a line connecting the points
( x1 , y1 ) and ( x2, y2 ) is equal to m = ( y2 – y1 ) /( x2 – x1 ).
Midpoint formula
The formula for finding the midpoint of the line connecting the points ( x1, y1 ) and ( x2, y2 ) is a new point whose abscissa is the average of the two given points’ x values, and whose ordinate is the average of the two given points’ y values. The midway is positioned exactly between the locations on the line that connects them.
[x y] = [x1+x2 /2 , y1+y2/2]
Coordinate Geometry Section Formula
The section formula is useful for determining the coordinates of a point on a line segment that splits the line segment joining the points ( x1, y1 ) and ( x2, y2 ) in the ratio m:n. The point that divides the provided two points is located on the line that connects them and can be found either The
[x y] = [mx1+ny2/m+n ;mx2+ny1/m+n]
Triangle’s centroid
The point of intersection of a triangle’s medians is the triangle’s centroid. (A median is a line that connects a triangle’s vertex to the opposite side’s midpoint.) The following formula is used to find the centroid of a triangle with vertices A ( x1, y1 ), B ( x2, y2 ), and C ( x3, y3 )
(x y) = (x1+x2+x3/3, y1+y2+y3/3)
Triangle Surface Area Formula for Coordinate Geometry
The area of a triangle with vertices A ( x1, y1 ), B ( x2, y2 ), and C ( x3, y3 ) is calculated using the formula below. This formula for calculating the area of a triangle can be applied to any triangle.
Area of a Triangle = 1 /2 | x1 ( y2– y3 ) + x2 ( y3 – y1 ) + x3 ( y1 – y2 ) |
Examples of coordinate geometry concepts
With the use of a basic linear equation, this line equation represents all of the points on the line. ax + by + c=0 is the conventional form of a line equation. There are several methods 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. The slope of the line is m, and the y-intercept of the line is c. The equation of a line also includes other types of line equations, such as point-slope form, two-point form, intercept form, and normal form.
Y = mx + c
The formula for Euclidean Distance
According to the Euclidean distance formula:
[(x2 – x1 )2+ (y2 – y1)2] = d
where,
The coordinates of one point are (x1, y1 ).
The coordinates of the other point are (x2, y2 ).
The distance between (x1, y1 ) and (x2, y2 ) is d.
Consider two places A (x1, y1 ) and B (x2, y2 ) and suppose that d is the distance between them to get the Euclidean distance formula. A line segment connects A and B. To find the formula, make a right-angled triangle with the hypotenuse AB. As seen below, we construct horizontal and vertical lines from A and B that intersect at C.
Consider two places A (x1, y1 ) and B (x2, y2 ) and suppose that d is the distance between them to get the Euclidean distance formula. A line segment connects A and B. To find the formula, make a right-angled triangle with the hypotenuse AB.
Conclusion
In this article, we have dealt with various concepts of coordinate geometry which will help us in better understanding. In this article, we have seen from the basic concept to complex theorems. We have seen all the formulas of coordinate geometry which will help us solve questions. We have also seen examples of coordinate geometry concepts.
