Coordinate Geometry
Every place on this planet has coordinates that help us to locate it easily on the world map. The coordinate system of our earth is made up of imaginary lines called latitudes and longitudes. The zero degrees 'Greenwich Longitude' and the zero degrees 'Equator Latitude' are the starting lines of this coordinate system. Similarly locating the point in a plane or a piece of paper, we have the coordinate axes with the horizontal xaxis and the vertical yaxis.
Coordinate geometry is the study of geometric figures by plotting them in the coordinate axes. Figures such as straight lines, curves, circles, ellipse, hyperbola, polygons, can be easily drawn and presented to scale in the coordinate axes. Further coordinate geometry helps to work algebraically and study the properties of geometric figures with the help of the coordinate system.
What Is Coordinate Geometry?
Coordinate geometry is an important branch of math, which helps in presenting the geometric figures in a twodimensional plane and to learn the properties of these figures. Here we shall try to know about the coordinate plane and the coordinates of a point, to gain an initial understanding of Coordinate geometry.
Coordinate Plane
A cartesian plane divides the plane space into two dimensions and is useful to easily locate the points. It is also referred to as the coordinate plane. The two axes of the coordinate plane are the horizontal xaxis and the vertical yaxis. These coordinate axes divide the plane into four quadrants, and the point of intersection of these axes is the origin (0, 0). Further, any point in the coordinate plane is referred to by a point (x, y), where the x value is the position of the point with reference to the xaxis, and the y value is the position of the point with reference to the yaxis.
The properties of the point represented in the four quadrants of the coordinate plane are:
 The origin O is the point of intersection of the xaxis and the yaxis and has the coordinates (0, 0).
 The xaxis to the right of the origin O is the positive xaxis and to the left of the origin, O is the negative xaxis. Also, the yaxis above the origin O is the positive yaxis, and below the origin O is the negative yaxis.
 The point represented in the first quadrant (x, y) has both positive values and is plotted with reference to the positive xaxis and the positive yaxis.
 The point represented in the second quadrant is (x, y) is plotted with reference to the negative xaxis and positive yaxis.
 The point represented in the third quadrant (x, y) is plotted with reference to the negative xaxis and negative yaxis.
 The point represented in the fourth quadrant (x, y) is plotted with reference to the positive xaxis and negative yaxis.
Coordinates of a Point
A coordinate is an address, which helps to locate a point in space. For a twodimensional space, the coordinates of a point are (x, y). Here let us take note of these two important terms.
 Abscissa: It is the x value in the point (x, y), and is the distance of this point along the xaxis, from the origin
 Ordinate: It is the y value in the point (x, y)., and is the perpendicular distance of the point from the xaxis, which is parallel to the yaxis.
The coordinates of a point are useful to perform numerous operations of finding distance, midpoint, the slope of a line, equation of a line.
Topics Covered in Coordinate Geometry
The topics covered in coordinate geometry helps in the initial understanding of the concepts and formulas required for coordinate geometry. The topics covered in coordinate geometry are as follows.
 About the Coordinate plane and the terms related to the coordinate plane.
 Know about the coordinates of a point and how the point is written in different quadrants.
 Formula to find the distance between two points in the coordinate plane.
 The formula to find the slope of a line joining two points.
 Midpoint Formula to find the midpoint of the line joining two points.
 Section Formula to find the points dividing the join of two points in a ratio.
 The centroid of a triangle with the given three points in the coordinate plane.
 Area of a triangle having three vertices in the coordinate geometry plane
 Equation of a line and the different forms of equations of a line
Coordinate Geometry Formulas
The formulas of coordinate geometry help in conveniently proving the various properties of lines and figures represented in the coordinate axes. The formulas of coordinate geometry are the distance formula, slope formula, midpoint formula, section formula, and the equation of a line. Let us know more about each of the formulas in the below paragraphs.
Coordinate Geometry Distance Formula
The distance between two points \((x_1, y_1)\) and \(x_2, y_2) \) is equal to the square root of the sum of the squares of the difference of the x coordinates and the ycoordinates of the two given points. The formula for the distance between two points is as follows.
D = \( \sqrt{(x_2  x_1)^2 + (y_2  y_1)^2}\)
Slope Formula
The slope of a line is the inclination of the line. The slope can be calculated from the angle made by the line with the positive xaxis, or by taking any two points on the line. The slope of a line inclined at an angle θ with the positive xaxis is m = Tanθ. The slope of a line joining the two points \((x_1, y_1)\) and \(x_2, y_2) \) is equal to m = \( \frac {(y_2  y_1)}{(x_2  x_1)} \).
m = Tanθ
m = \((y_2  y_1)\)/\((x_2  x_1)\)
MidPoint Formula
The formula to find the midpoint of the line joining the points \((x_1, y_1)\) and \(x_2, y_2) \) is a new point, whose abscissa is the average of the x values of the two given points, and the ordinate is the average of the y values of the two given points. The midpoint lies on the line joining the two points and is located exactly between the two points.
\((x, y) =\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)\)
Section Formula in Coordinate Geometry
The section formula is useful to find the coordinates of a point that divides the line segment joining the points \((x_1, y_1)\) and \((x_2, y_2)\) in the ratio \(m : n\). The point dividing the given two points lies on the line joining the two points and is available either between the two points or outside the line segment between the points.
\((x, y) = \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right) \)
The centroid of a Triangle
The centroid of a triangle is the point of intersection of medians of a triangle. (Median is a line joining the vertex of a triangle to the midpoint of the opposite side.). The centroid of a triangle having its vertices A\((x_1, y_1)\), B\((x_2, y_2)\), and C\((x_3, y_3)\) is obtained from the following formula.
\((x, y) = (\dfrac{x_1+ x_2 + x_3}{3}, \dfrac{y_1 + y_2 + y_3}{3})\)
Area of a Triangle Coordinate Geometry Formula
The area of a triangle having the vertices A\((x_1, y_1)\), B\((x_2, y_2)\), and C\((x_3, y_3)\) is obtained from the following formula. This formula to find the area of a triangle can be used for all types of triangles.
Area of a Triangle = \(\dfrac{1}{2}x_1(y_2  y_3) + x_2(y_3  y_1) + x_3(y_1  y_2)\)
How to Find Equation of a Line in Coordinate Geometry?
This equation of a line represents all the points on the line, with the help of a simple linear equation. The standard form of the equation of a line is ax + by + c= 0. There are different methods to find the equation of a line. Another important form of the equation of a line is the slopeintercept form of the equation of a line (y = mx + c). Here m is the slope of the line and c is the yintercept of the line. Further, the other forms of equations of a line such as pointslope form, twopoint form, intercept form, and the normal form, are presented in the equation of a line webpage of cuemath.
y = mx + c
Related Topics to Coordinate Geometry
 Cartesian System
 Cartesian Coordinates
 Distance Formula
 Distance Between Two Points
 Midpoint
 Slope
 Midpoint Formula
 Equation of a Line
 Three Dimensional Distance Formula
 Distance of a Point From a Line
 SlopeIntercept Form of a Line
 Point Slope Form
 Euclidean Distance Formula
Tips And Tricks on Coordinate Geometry
 The slope of the xaxis is 0 and the slope of the yaxis is \(\infty\).
 The equation of xaxis is y = 0 and the equation of yaxis is x = 0
 A point on the \(x\)axis is of the form (a, 0), and a point on the yaxis is of the form (0, b)
 Point Slope Form of equation of a line is \((y  y_1) = m(x  x_1) \).
 Two Point Form of equation of a line is \(y  y_1 = \left(\dfrac{y_2  y_1}{x_2  x_1}\right).(x  x_1) \)
 The slope Intercept Form of the equation of a line is y = mx + c
 For two parallel lines in the coordinate plane, their slopes are equal.
 And for two perpendicular lines in the coordinate plane, the product of the slopes is equal to 1.
Solved Examples on Coordinate Geometry

Example 1: Ron is given the coordinates of one end of the diameter of a circle as (5, 6) and the center of the circle as (2, 1). Using the formulas of coordinate geometry how can we help Ron to find the other end of the diameter of the circle?
Solution:
Let \(AB\) be the diameter of the circle with the coordinates of points \(A \), and \(B\) as follows.
\( A = (x_1, y_1) \), \(B = (x_2, y_2) = (5, 6)\)
The coordinates of the center \(O = (x, y) = (2, 1)\)
The coordinate geometry formula for midpoint of the line is:
\[ (x, y) = \left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right) \]
Applying this we have the following calculations.
\[\begin{align} (2, 1) &=\left (\frac{x_1 + 5}{2}, \frac{y_1 + 6}{2}\right) \end{align} \]
Here we shall segregate the coordinates and the \(x\) value is:
\[\begin{align} \dfrac{x_1 + 5}{2} &= 2 \\x_1 + 5 &= 2 \times 2\\x_1 + 5 &=4 \\ x_1 &=4 5 \\x_1 &= 9 \end{align} \]
And the \(y\) value is:
\[\begin{align} \dfrac{y_1 + 6}{2} &= 1 \\y_1 + 6&= 1 \times 2\\y_1 + 6 &=2 \\ y_1 &=2  6 \\y_1 &= 4 \end{align} \]
Therefore the point \(A = (x_1, y_1) = (9, 4)\)
Answer: Therefore the other end of the diameter is (9, 4).

Example 2: Find the equation of a line passing through (2, 3) and having a slope of 1.
Solution:
The point on the line is \((x_1, y_1) = (2, 3)\), and the slope is \(m = 1\).
Using the coordinate geometry point and slope form of the equation of the line, we have:
\[\begin{align}(y  y_1) &= m(x  x_1) \\ (y  3) &=(1)(x (2)) \\ y  3 &= (x + 2) \\ y  3 &= x 2 \\ x + y &= 3  2 \\ x + y &= 1\end{align} \]
Answer: Therefore the equation of the line is x + y = 1.

Example 3: Find the equation of a line having a slope of 2 and \(y\)intercept of 1.
Solution:
The given information is \(m = 2\) and \(y\)intercept is \( c = 1\)
From coordinate geometry we can use the slope intercept form of equation of a line.
\[\begin{align} y &= mx + c \\ y &= (2)x + 1 \\ y &= 2x + 1 \\ 2x + y &= 1\end{align} \]
Answer: Therefore the equation of the line is 2x + y = 1.
FAQs on Coordinate Geometry
What Is Coordinate Geometry?
Coordinate Geometry is helpful to define the points in space. For this, the primary axis of the xaxis and yaxis is defined and then the points are measured and marked with reference to these points. Further, the various geometric figures such a line, curve, circle, ellipse, hyperbola, can be plotted in the coordinate axes and we can study the various properties of these geometric figures.
What Is Distance Formula in Coordinate Geometry?
The distance formula is useful to find the distance between two points in a coordinate plane. For points \((x_1, y_1)\) and \((x_2, y_2)\), the formula to find the distance is D = \(\sqrt{(x_2  x_1)^2 + (y_2  y_1)^2} \).
What is Slope in Coordinate Geometry?
The slope of a line can be found in two ways in coordinate geometry. For the given angle of inclination θ of the line with the positive xaxis, the slope of the line is m = Tanθ. For the given two points \((x_1, y_1)\) and \((x_2, y_2)\), on the line, the slope of the line is equal to m = \(\dfrac{(y_2  y_1)}{(x_2  x_1)}\).
What Are Collinear Points in Coordinate Geometry?
The collinear points in coordinate geometry refer to a set of points that lie on the same line. The condition for three points to be collinear is that the largest distance between two points is equal to the sum of the distances between the other two sets of points. Also, the collinear points can be found using the slope formula. The slope of the line joining two points should be equal to the slope of the line joining the other two points.
Where Is Coordinate Geometry Used in Maths?
The concepts of coordinate geometry have wide applications in math. The topis of maths such as vectors, threedimensional geometry, equations, calculus, complex numbers, functions have numerous applications of coordinate geometry. All of these topics require the data to be graphically presented in a two/threedimensional coordinate plane.
What is Section Formula in Coordinate Geometry?
The section formula is useful to find the coordinates of a point which divides the line segment joining the points \((x_1, y_1)\) and \((x_2, y_2)\) in the ratio \(m : n\). The point dividing the line segment lies on the line joining the two points, and is either present between the two points or is beyond the two points. The formula to find the required point is: \((x, y) = \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right) \)
How to Find the Area of Triangle in Coordinate Geometry?
The area of a triangle joining the three points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) in the coordinate system is \( \frac {1}{2}.x_1(y_2  y_3) + x_2(y_3  y_1) + x_3(y_1  y_2)\). The modulus symbol is used in the formula, since the area is always a positive value.
How Is Coordinate Geometry Used in Real Life?
There are numerous applications of coordinate geometry in our real life. The maps we use to locate places: google maps, physical maps, are all based on the coordinate system. Further, it is helpful in largescale land projects to draw the land maps to scale. The naval engineers use coordinate systems, to locate any point in the seas.