# Coordinate Geometry

(Last Updated On: March 24, 2023)

Coordinate geometry is a branch of mathematics that deals with the study of geometric shapes and figures on a coordinate plane. In this system, points in a plane are represented by ordered pairs of numbers, called coordinates, which determine the position of the points relative to a fixed origin. The coordinates are used to specify the position of points, lines, and curves in the plane. The study of coordinate geometry involves finding the equations of lines and curves, determining the intersections and distances between points, and studying the geometric properties of figures, such as the midpoint and slope of a line, the distance formula, and the equation of a circle. Coordinate geometry is widely used in various fields, including engineering, physics, and computer graphics, among others.

## What Is Coordinate Geometry?

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Coordinate Geometry is a branch of mathematics that deals with the study of points, lines, and planes in a two-dimensional space or a three-dimensional space. It is based on the idea of assigning each point in the space a set of numerical coordinates that uniquely determine its position.

In two-dimensional coordinate geometry, points are represented by ordered pairs of numbers (x, y) and are graphed on a coordinate plane. The x-coordinate determines the position of the point along the horizontal axis.

### What is Coordinate Plane with example

A coordinate plane is a two-dimensional graph that is used to represent mathematical concepts and relationships. It is comprised of two perpendicular lines, the x-axis and y-axis, which intersect at a fixed point called the origin. The x-axis is horizontal and the y-axis is vertical.

In a coordinate plane, each point can be represented by an ordered pair of numbers (x, y), where x represents the point’s position along the x-axis and y represents its position along the y-axis. The origin, for example, is represented by the point (0, 0).

Let’s consider a simple example to help understand the coordinate plane. Suppose we want to plot the point (2, 3) in a coordinate plane. We start by drawing the x-axis and y-axis, then count 2 units to the right along the x-axis from the origin and 3 units up along the y-axis. The point where the x and y axes intersect is the point (2, 3).

This is a simple example, but the coordinate plane can be used to represent much more complex mathematical concepts and relationships. By plotting points and lines in a coordinate plane, we can visualize and study geometric shapes, curves, and other mathematical entities in a convenient and intuitive way.

### Coordinates of a point

The coordinates of a point in a coordinate plane are a set of two numbers that describe the position of the point relative to the origin. In a two-dimensional coordinate plane, the origin is represented by the point (0, 0) and the coordinates of a point are represented by an ordered pair (x, y). The first number x represents the point’s position along the horizontal axis (the x-axis), and the second number y represents its position along the vertical axis (the y-axis).

For example, the point (3, 4) has an x-coordinate of 3, which means that it is located 3 units to the right of the origin along the x-axis. The y-coordinate of 4 means that the point is located 4 units above the origin along the y-axis.

The coordinates of a point can also be used to plot the point on the coordinate plane. To plot a point with coordinates (x, y), we start at the origin and move x units to the right along the x-axis, and then move y units up along the y-axis. The point where the x and y axes intersect is the point (x, y).

## Coordinate Geometry Formula

### Coordinate Geometry Distance Formula

The distance formula in coordinate geometry is used to find the distance between two points in a plane. The formula is based on the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

In a coordinate plane, the distance between two points (x1, y1) and (x2, y2) can be found using the following formula:

d = √((x2 – x1)^2 + (y2 – y1)^2)

where d is the distance between the two points and (x1, y1) and (x2, y2) are the coordinates of the two points.

The formula is based on the idea that the line connecting two points in the coordinate plane can be thought of as the hypotenuse of a right triangle. The distance between the two points is then equivalent to the length of the hypotenuse. By using the coordinates of the two points, we can calculate the lengths of the other two sides of the right triangle and apply the Pythagorean theorem to find the distance.

The distance formula is widely used in geometry, physics, and engineering, among other fields, to find distances, calculate distances between points, and study geometric properties of figures.

### Slope Formula

The slope formula is a mathematical expression that is used to calculate the slope of a line in a coordinate plane. The slope of a line is a measure of its steepness and direction, and is represented by the letter m. It is defined as the ratio of the vertical change (rise) between two points on the line to the horizontal change (run) between the same two points.

The slope formula can be derived from the coordinates of two points (x1, y1) and (x2, y2) on the line, and is given by:

m = (y2 – y1) / (x2 – x1)

where m is the slope of the line, (x1, y1) and (x2, y2) are the coordinates of two points on the line, and (x2 – x1) is the horizontal change (run) between the two points.

The slope formula is a powerful tool that can be used to find the equation of a line in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept (the point where the line crosses the y-axis). It can also be used to determine the angle of a line, to classify lines as horizontal, vertical, or diagonal, and to find the slope of a perpendicular or parallel line. The slope formula is widely used in mathematics, engineering, and physics, among other fields.

### Mid-Point Formula

The midpoint formula is a mathematical expression used to find the midpoint of a line segment in a coordinate plane. The midpoint of a line segment is the point that divides the line segment into two equal parts.

The midpoint formula can be derived from the coordinates of the two endpoints of the line segment, (x1, y1) and (x2, y2), and is given by:

Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

where (x1, y1) and (x2, y2) are the coordinates of the two endpoints of the line segment, and the midpoint is represented by the ordered pair ((x1 + x2) / 2, (y1 + y2) / 2).

The midpoint formula is a simple and useful tool for finding the center of a line segment, and for dividing a line segment into equal parts. It is widely used in geometry, trigonometry, and other mathematical disciplines. By finding the midpoint of a line segment, we can create new lines and shapes, or use it as a reference point to study properties of the original line segment.

### Section Formula in Coordinate Geometry

The section formula in coordinate geometry is a mathematical expression used to find a point on a line segment that is a specified fraction of the distance between the endpoints of the line segment. The section formula is commonly used to find the midpoint of a line segment, as well as other points along the line segment that are a specific fraction of the distance between the endpoints.

The section formula can be derived from the coordinates of the two endpoints of the line segment, (x1, y1) and (x2, y2), and a fraction, t, which is a value between 0 and 1. The fraction t represents the fraction of the distance between the two endpoints that we want to find. The formula is given by:

P = (1 – t) * (x1, y1) + t * (x2, y2) = (x1 + t * (x2 – x1), y1 + t * (y2 – y1))

where P = (x, y) is the point on the line segment that is a fraction t of the distance between the endpoints, and (x1, y1) and (x2, y2) are the coordinates of the endpoints of the line segment.

The section formula can be used to find the midpoint of a line segment by setting t = 0.5. It can also be used to find any other point along the line segment by using a different value of t. The section formula is widely used in geometry, trigonometry, and other mathematical disciplines, and is a useful tool for finding points along a line segment and for studying the properties of the line segment.

### The Centroid of a Triangle

The Centroid of a triangle is a point that is the arithmetic mean (average) of the three vertices of the triangle. It is also known as the “center of mass” of the triangle, and is denoted by the letter G. The centroid is always located inside the triangle and is equidistant from each of the three vertices.

The coordinates of the centroid can be found by averaging the x-coordinates and y-coordinates of the vertices of the triangle. Let (x1, y1), (x2, y2), and (x3, y3) be the vertices of the triangle, then the coordinates of the centroid are given by:

G = ((x1 + x2 + x3) / 3, (y1 + y2 + y3) / 3)

The centroid is an important point in geometry, as it has several useful properties. For example, the medians of a triangle (lines that connect each vertex to the midpoint of the opposite side) intersect at the centroid, and the centroid divides each median into two segments in the ratio 2:1, with the longer segment being twice the length of the shorter segment. The centroid is also a center of gravity for the triangle, meaning that if the triangle were cut out of a uniform material and hung from the centroid, it would remain in balance.

In addition, the centroid is used in many applications, such as in engineering and physics, where it is used to find the center of mass of a triangle-shaped object. The centroid is also useful in computer graphics and image processing, where it is used to find the center of a triangle for various transformations.

### Area of a Triangle Coordinate Geometry Formula

The area of a triangle in coordinate geometry can be found using the coordinates of its vertices. The formula for finding the area of a triangle in coordinate geometry is based on the Shoelace theorem.

The Shoelace theorem states that the area of a polygon with vertices (x1, y1), (x2, y2), …, (xn, yn) can be found by taking half of the absolute value of the sum of the product of the x-coordinates and y-coordinates of consecutive vertices, after rearranging them in a specific order.

For a triangle with vertices (x1, y1), (x2, y2), and (x3, y3), the formula for finding the area is given by:

Area = | (x1 * y2 + x2 * y3 + x3 * y1) – (x2 * y1 + x3 * y2 + x1 * y3) | / 2

where the absolute value is taken to account for the possibility of negative areas.

The Shoelace theorem is a simple and elegant method for finding the area of a triangle in coordinate geometry, and can also be used to find the area of other polyggonal shapes by simply adding additional vertices. The Shoelace theorem is widely used in computer graphics, image processing, and other applications where the area of a polygon needs to be found from its vertices.

### How to Find Equation of a Line in Coordinate Geometry?

There are several methods to find the equation of a line in coordinate geometry, but two of the most common are the slope-intercept form and the point-slope form.

1. Slope-intercept form: This form is y = mx + b, where m is the slope of the line and b is the y-intercept. To find the equation of a line in this form, you need to know the slope and one point on the line.

Suppose you have a line with slope m and a point (x1, y1) on the line. Then the equation of the line can be found using the following steps:

• Substitute the slope and the point into the slope-intercept form: y = mx + b
• Solve for b: b = y1 – mx1
• Substitute the values of m and b back into the equation: y = mx + (y1 – mx1)
1. Point-slope form: This form is y – y1 = m(x – x1), where m is the slope of the line and (x1, y1) is any point on the line. To find the equation of a line in this form, you need to know the slope and one point on the line.

Suppose you have a line with slope m and a point (x1, y1) on the line. Then the equation of the line can be found using the following steps:

• Substitute the slope and the point into the point-slope form: y – y1 = m(x – x1)
• Simplify the equation: y = mx – mx1 + y1

Both the slope-intercept form and the point-slope form are useful for finding the equation of a line in coordinate geometry, and can be used to graph the line, find its slope and y-intercept, and solve problems involving lines.

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?

We can use the formula for the distance between two points in coordinate geometry to help Ron find the other end of the diameter of the circle.

The formula for the distance between two points (x1, y1) and (x2, y2) is given by:

d = √((x2 – x1)^2 + (y2 – y1)^2)

In this problem, one end of the diameter of the circle is given as (5, 6), and the center of the circle is given as (-2, 1). Let’s call the other end of the diameter (x, y).

Using the formula for the distance between two points, we have:

d = √((x – 5)^2 + (y – 6)^2) = √((-2 – 5)^2 + (1 – 6)^2)

Expanding the squares and solving for x and y, we find:

d^2 = (x – 5)^2 + (y – 6)^2 = (7)^2 + (5)^2 x – 5 = ±7 y – 6 = ±5 x = 5 ± 7 y = 6 ± 5

So the other end of the diameter could be either (12, 11) or (-2, 1).

Note that in this case, the center of the circle is also one of the ends of the diameter, so there are two possible answers

Find the equation of a line passing through (-2, 3) and having a slope of -1.

The equation of a line in point-slope form is given by y – y1 = m(x – x1), where (x1, y1) is a point on the line and m is the slope of the line.

Given the coordinates of a point (-2, 3) on the line and the slope of the line (-1), we can find the equation of the line as follows:

• Substitute the coordinates of the point and the slope into the point-slope form: y – 3 = -1(x – (-2))
• Simplify the equation: y – 3 = -1(x + 2)
• Distribute the -1: y – 3 = -x – 2
• Add x to both sides: y – 3 + 3 = -x
• Add x to both sides: y = -x + 5

So the equation of the line passing through (-2, 3) and having a slope of -1 is y = -x + 5.

NCERT Solutions for Class 9 Maths Chapter 1 Number Systems Ex 1.2

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