What is Coordinate Geometry? 

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates. 

It is possible to find the distance between two points, divide lines in m:n ratios, find the mid-point of a line, calculate the area of a triangle in the Cartesian plane, and so on using coordinate geometry. 

The Coordinate Plane 

The coordinate plane is the set of all points in coordinate geometry. 

The x-axis, which runs through the plane, and the y-axis, which is at right angles to the x-axis, are the two scales in the diagram above. On the x-axis, point A has a value of 20 and on the y-axis, it has a value of 15. These are also the coordinates for point A. 

• The origin is defined as the point where the x and y axes intersect. The coordinates of the origin are (0, 0). 
• On the right-hand side of the x-axis, the values are always positive and the values on the left-hand side of the x-axis are negative. 
• Similarly, positive values appear above the origin on the y-axis, while negative values appear below the origin. 
• The names of horizontal and vertical lines that are drawn to determine the position of a point in the Cartesian plane are known as the x-axis and y-axis. 

As you can see, the axes divide the plane into four parts. In anticlockwise order, the quadrants (fourth part) are numbered I, II, III, and IV. 

The axes and these quadrants make up the plane. The Cartesian plane, coordinate plane, or xy-plane are all terminology for the same plane. The axes are called the coordinate axes. 

The following conventions can be used to express the coordinates of a point. 

• The x-coordinate of a point is the perpendicular distance from the y-axis determined along the x-axis. The abscissa is another name for the x-coordinate. 
• The y-coordinate of a point is the perpendicular distance from the x-axis determined along the y-axis. The ordinate is another name for the y-coordinate. 
•  When listing the coordinates of a point in the coordinate plane, the x-coordinate comes first, followed by the y-coordinate. The coordinates are enclosed in parentheses. 

Since the distance of the point (2,3) from the y-axis is 2 units, the x-coordinate or abscissa of the point is 2. The point (2,3) is three units away from the x-axis therefore, the y-coordinate, i.e., the ordinate, of the point is 3. As a result, the point's coordinates are (2, 3). 

Straight Line 

A line is a geometrical object that is straight, infinitely long, and infinitely thin. Two or more points on a line whose coordinates are known, determine its position. A straight line in a plane is a set of points where all of the points' coordinates satisfy a given linear equation, and this linear equation is not satisfied by coordinates of any other point on the plane that does not lie on the line. 

The simplest geometrical figure is the straight line. A straight line is the shortest distance between any two points on a plane. 

A vertical line is a line that runs up and down the page in geometry. If any two points on a line have the same x-coordinate, the line is vertical. The horizontal line is a line that runs through the page from left to right.  If any two points on a line have the same y-coordinate, the line is horizontal. A vertical and a horizontal line are always perpendicular. 

Distance Formula 

If we know the coordinates of two points, we can use the distance formula to calculate the distance between them. These coordinates may be on the x, y, or both axes. Assume P and Q are two points in an XY plane. The coordinates of point P are (x₁,y₁), and the coordinates of point Q are (x₂,y₂). The following is the formula for measuring the distance between two points PQ: 

The formula is, 

PQ=√[(x₂-x₁)²+(y₂-y₁)²] 

 The Pythagorean Theorem is used to derive the distance formula to measure the distance between any two points. Typically, these points are generated on the xy-coordinate plane. 

In geometry, the hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. From the above figure, segment AB is the hypotenuse. 

The midpoint of Line Segment 

A midpoint is the point on a line segment that splits it into two equal parts. A midpoint can only be found in a line segment. Since a line extends in both directions indefinitely, it cannot have a midpoint. A ray can't have a midpoint because it only has one end.  

The midpoint of the segment joining the points (x₁,y₁) & (x₂,y₂) is given by:  

Midpoint = $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

The slope of a Line 

A line's slope is a number that defines its "steepness," usually represented by the letter m. It's the change in y for a unit change in x along the line. The line's slope is constantly recalculated. 

The slope of a line is calculated by measuring the rise (the amount that y changes) over the run (how much x changes). 

The formula for slope is: m = [y₂ - y₁] / [x₂ - x₁] 

Equation of Line 

When written in the "slope and intercept" form, the equation of a line is y = mx+b. The slope of the line defined here is represented by ‘m’ in the equation. 

Intercept of Line 

The y-value of a line's intersection with the y-axis is its intercept. The point where a line crosses the x or y-axis is called the intercept. The y-axis is assumed if we don't specify which one. The letter b is commonly used to mark it. 

The intercept (b) is given by: 

b=y-mx 

Where the slope of the line is m. 

Coordinate Geometry Figures 

Triangle 

A triangle is a polygon with three sides and three vertices in geometry. A triangle's area is the amount of space it takes up in a two-dimensional plane. If the vertices of a triangle are known, we must first determine the length of the triangle's three sides. The distance formula can be used to calculate the length. Let us assume a triangle PQR, whose coordinates P, Q, and R are given as (x₁, y₁), (x₂, y₂), (x₃, y₃), respectively. 

Formula for the area of a triangle: A = (1/2) |x₁ (y₂ – y₃ ) + x₂ (y₃– y₁ ) + x₃(y₁– y₂)| 

Isosceles Triangle 

Two congruent sides and two congruent angles make up an isosceles triangle. The sides are the simplest way to prove that a triangle is isosceles using coordinate geometry. Distance formula can be used to calculate the side length of the triangle. 

Parallelogram 

A parallelogram is similar to an ordinary parallelogram in coordinate geometry, with the exception that its coordinate plane location is known. The coordinates of each of the four vertices (corners) are known. Various properties, such as the altitude, can be determined using these coordinates. The area of a parallelogram is the altitude times the base. 

Conic Sections 

When a plane intersects a right circular double cone, conic sections are created. Depending on the angle created by the plane with the cone's vertical axis and the direction of the intersecting plane with respect to the cone, we can obtain various types of conic sections. The plane can intersect the cone at any point on its surface, including the vertex. We get a circle, ellipse, parabola, or other shapes when the plane cuts the cone's nappe (other than the vertex). 

Formulas 

Distance formula: $\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$, where $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ are two points between which the distance is calculated.

Midpoint formula: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$, where $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ are two end-points of the segment.

The slope of a line: $\frac{y_2-y_1}{x_2-x_1}$, where $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ are any two points on the line.

Slope-intercept form for equation of a line: y=mx+c, where m is the slope and c is the y-intercept of the line. 

Area of a triangle: $\frac{1}{2}\left|x_1 \left(y_2 –y_3 \right)+x_2 \left(y_{3 }–y_1 \right)+x_3\left(y_{1 }–y_2\right)\right|$, where $\left(x_1,y_1\right)$, $\left(x_2,y_2\right)$ and $\left(x_3,y_3\right)$ are the vertices of the triangle.

Common Mistakes 

• The lengths are generally rounded to one decimal position and the coordinates are rounded to integers. Calculations could be slightly off as a result of this. 
• Error while writing the ratio of corresponding sides. 

Context & Applications  

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for 

• 10th,11th,12th Standard. 
• Engineering Students. 
• To locate the location of the cursor or finger in digital devices such as computers, cell phones, and so on. 
• In navigation and on maps (GPS). 
• In astronomy, coordinate geometry is used to compute the movements of celestial bodies such as planets, comets, binary star systems, and so on. 

Related Concepts 

• Conic sections 
• Circles, ellipse, hyperbola