What Is Coordinate Geometry
Coordinate geometry is similar to pure geometry in that it focuses on objects like points, lines, and circles. Unlike pure geometry, however, it uses a reference system and units to define properties of these objects.
For example, in pure geometry, a point is simply that which has no part, and its existence will be postulated. In coordinate geometry, on the other hand, the location of a point relative to other points or objects is just as important as its existence.
Because coordinate geometry uses units, it is possible to develop equations and formulae to relate objects and discover properties about objects. Some common examples include distance, area, and circumference.
Coordinate Geometry In Two Dimensions
Unless otherwise specified, coordinate geometry usually refers to two-dimensional coordinate geometry. The most common coordinate system used is the Cartesian coordinate system, which is sometimes called rectangular coordinates.
The Cartesian coordinate system has a horizontal axis called the x-axis and a vertical axis called the y-axis. These two axes meet at the origin. The expression references a point in this system. Here, x is the horizontal distance from the origin and y is the vertical distance from the origin. A negative number signifies leftward or downward movement. On the other hand, a positive number specifies rightward or upward movement. The origin has coordinates , while the point A in the image below has coordinates .
Plotting A Point In The Coordinate Plane
In this section, we are going to learn how to plot a point on the coordinate plane. Let’s take the example of point P = . To plot a point in the coordinate plane, follow the steps given below:
- Step 1: Draw two perpendiculars, the X-axis and Y-axis.
- Step 2: Start from the origin. Move 5 units to the right, along the positive X-axis.
- Step 3: Move 6 units up, along the positive Y-axis.
- Step 4: Mark the point of intersection. Mark it as .
Note that P is in the first quadrant. Also, this is known as the positive coordinate plane as the value of both the coordinates for any point in this quadrant will be positive.
Important Points on Coordinate Plane:
- The first quadrant known as the positive coordinates quadrant, is represented by the Roman numeral I.
- The second quadrant is represented by the Roman numeral II.
- The third quadrant is represented by the Roman numeral III.
- The fourth quadrant is represented by the Roman numeral IV.
- The coordinates of any point are enclosed in brackets.
Try to Solve this Challenging Question:
Find out any three points that lie in the positive coordinate plane and for which the abscissa and ordinate are equal and non-negative.
Topics Related to Coordinate Plane
Example 1: Let’s help Olivia and Jane plot the following points in the Cartesian plane:
A and C are in the first quadrant.B is in the second quadrant.D is in the fourth quadrant.
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Angle Formula: To Find The Angle Between Two Lines
Consider two lines A and B, having their slopes to be \ respectively.
Let be the angle between these two lines, then the angle between them can be represented as-
Special Cases:
- Case 1: When the two lines are parallel to each other,
\ = m
Substituting the value in the equation above,
- Case 2: When the two lines are perpendicular to each other,
m1 . m2 = -1
Substituting the value in the original equation,
\ which is undefined.
= 90°
Locating A Point On The Coordinate Plane
Now that we are already familiar with the coordinate plane and its parts, let’s discuss how to identify points on a coordinate plane. To locate a point on the coordinate plane, follow the steps given below:
- Step 1: Locate the point.
- Step 2: Find the quadrant by looking at the signs of its X and Y coordinates.
- Step 3: Find the X-coordinate or abscissa of the point by reading the number of units the point is to the right/left of the origin along the X-axis.
- Step 4: Find the Y-coordinate or the ordinate of the point by reading the number of units the point is above/below the origin along the Y-axis.
Let’s look at the coordinate plane examples. Look at the figure shown below.
- Step 1: Observe the blue dot on the coordinate graph.
- Step 2: It is in the second quadrant.
- Step 3: The point is 3 units away from the origin along the negative X-axis.
- Step 4: The point is 2 units away from the origin along the positive Y-axis.
Thus, the point on the graph has coordinates .
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Geometry In Everyday Life
Geometry was thoroughly organized in about 300bc, when the Greek mathematician, Euclid gathered what was known at the time added original work of his own and arranged 465 propositions into 13 books, called Elements. Geometry was recognized to be not just for mathematicians. Anyone can benefit from the basic learning of geometry, which is to follow the lines reasoning. Geometry is one of the oldest sciences and is concerned with questions of shape, size and relative position of figures and with properties of space. Geometry is considered an important field of study because of its applications in daily life. Geometry is mainly divided in two
Plane geometry It is about all kinds of two dimensional shapes such as lines,circles and triangles.
Solid geometry It is about all kinds of three dimensional shapes like polygons,prisms,pyramids,sphere and cylinder. Role of geometry in daily life Role of geometry in the daily life is the foundation of physical mathematics. A room, a car, a ball anything with physical things is geometrically formed. Geometry applies us to accurately calculate physical spaces. In the world , Anything made use of geometrical constraints this is important application in daily life of geometry.
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Using Coordinate Planes For Other Problems
You can also use coordinate planes in a bit more of an abstract way, to describe how one quantity varies with another. By labeling your independent variable x and your dependent variable y, you can use a coordinate plane to describe pretty much any relationship. For example, if your independent variable is the price of an item and the dependent variable is how many of them you sell, you can create a graph in the coordinate plane to help you understand the relationship. You can apply this to a huge range of different problems, because the coordinate plane allows you to see how one quantity varies with another in a visual way.
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Finding Intersections Of Geometric Objects
For two geometric objects P and Q represented by the relations P the intersection is the collection of all points ( which are in both relations.
For example, might be the circle with radius 1 and center ( +y^=1\}} and might be the circle with radius 1 and center ( } +y^=1\}} . The intersection of these two circles is the collection of points which make both equations true. Does the point ( make both equations true? Using ( +0^=1} or . On the other hand, still using ( +0^=1} or so it is not in the intersection.
The intersection of can be found by solving the simultaneous equations:
- x +y^=1}
- ( +y^=1.}
Traditional methods for finding intersections include substitution and elimination.
Substitution: Solve the first equation for y and then substitute the expression for y
- +y^=1}
- y =1-x^} .
We then substitute this value for y into the other equation and proceed to solve for x
- +=1}
- x -2x+1+1-x^=1}
Next, we place this value of x in either of the original equations and solve for y
- +y^=1}
- y
- }}}.}
So our intersection has two points:
- \ \ \mathrm \ \ \left.}
Elimination: Add a multiple of one equation to the other equation so that one of the variables is eliminated. For our current example, if we subtract the first equation from the second we get ( -x^=0} . The in the first equation is subtracted from the y in the second equation leaving no y has been eliminated. We then solve the remaining equation for x , in the same way as in the substitution method:
- x -2x+1+1-x^=1}
- +y^=1}
- y
- }}}.}
Coordinate Graphing Of Real World Problems
Patterns are all around you. From the six-pack of sodas to the dozen eggs you might buy, there is something to notice everywhere you look.
You may not have realized it before, but your powers of observation are a mathematical skill. By graphing your observations on a coordinate graph, you can make a visual picture of the relationships you observe between numbers.
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Reading Or Plotting Coordinates In Graph
We always write coordinates in brackets, the two coordinates are separated by comma. As coordinates are ordered pairs of numbers the first number represents the point on the \-axis and the second represents the point on the \-axis.When reading or plotting coordinates, we always go across first and then up. A good way to remember this is: Across the landing and up the stairs. To plot the points \ in the Cartesian coordinate plane, we follow the \-axis until we reach \ and draw a vertical line at \.Similarly, we follow the \-axis until we reach \ and draw a horizontal line at \. The intersection of these two lines is the position of \ in the Cartesian plane. This point is at a distance of \ units from the \-axis and \ units from the \-axis. Thus, the position of \) is located in the cartesian plane.
What Is A Coordinate Plane
A coordinate plane is a two-dimensional surface formed by two number lines. It is formed when a horizontal line called the X-axis and a vertical line called the Y-axis intersect at a point called the origin. The numbers on a coordinate grid are used to locate points. A coordinate plane can be used to graph points, lines, and much more. It acts as a map and yields precise directions from one point to another.
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What Is The Use Of Coordinate Geometry In Daily Life
Coordinate Geometry illustrates the link between geometry and algebra through graphs connecting curves and lines. It gives geometric aspects in Algebra and enables them to solve geometric problems. It is a part of geometry where the position of points on the plane is described using an ordered pair of numbers
How Did Rectangular Coordinate System Or Cartesian Plane Helps U In Real Life Situation
A rectangular coordinate system, or a Cartesian coordinate system, can also be applied in daily life. On the y-axis, which is the horizontal line, we can measure the total amount of sales or how much cereal was sold, and then on the x-axis, which is the horizontal line, we can measure the date being measured.
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Applications Of Coordinate Geometry
Listed below are few applications of coordinate geometry
- It is used to find the distance between two points
- It is used to find the ratio of dividing lines in the m:n ratio
- It is used to find the mid-point of a line
- It is used to calculate the area of a triangle in the Cartesian plane
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Use Of Coordinate Geometry In Real Life
Understanding and learning coordinate geometry is an important math skill that also holds great significance in real life. The concept of coordinate geometry was first introduced in the 17th century by the French mathematician Rene Descartes. It is the branch of geometry that forms the link between algebra and geometry through lines and curves. It is also known as analytical geometry that involves the use of coordinate planes and coordinate points. A coordinate plane is a two-dimensional plane with an x-axis and y-axis. These x and y-axis intersect each other at a point called the origin of the coordinate plane. Coordinates are ordered pairs of numbers that define the value of x and y variables to determine the position of points on a coordinate plane.
Using coordinate geometry in real life means putting all these abstract concepts and terminologies into real-world situations. Thus gaining an in-depth understanding of these concepts is highly crucial for clearly implementing them practically. Cuemath online math resources offer interactive learning for students to learn various mathematical concepts and their applications quickly. You can easily find some of the coordinate geometry worksheets that help students to grasp these complex concepts with simple illustrations and exercises easily. Here are a few examples that will help students to visualize the use of coordinate geometry in real life:
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What Is The Distance Formula
The distance formula is the formula, which is used to find the distance between any two points, only if the coordinates are known to us. These coordinates could lie in x-axis or y-axis or both. Suppose, there are two points, say P and Q in an XY plane. The coordinates of point P are and of Q are . Then the formula to find the distance between two points PQ is given by:
PQ =
D = |
Where D is the distance between the points.
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.
- Mid-point 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
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Sign Convention In Cartesian Plane
A Cartesian plane is divided into four quadrants by two coordinate axes perpendicular to each other. The four quadrants, along with their respective sign convention of ordered pairs, are represented in the graph below.1. If a point is in the \ quadrant, then the ordered pair will be in the form \.2. If a point is in the \ quadrant, then the ordered pair will be in the form \.3. If a point is in the \ quadrant, then the ordered pair will be in the form \.4. If a point is in the \ quadrant, then the ordered pair will be in the form \.
The figure given below shows the coordinates \ lies in the \ quadrant since both the coordinates are positive.
The \-coordinate of a point is its perpendicular distance from the \-axis measured along the \-axis -axis and negative along the negative direction of the \-axis). The \-coordinate is also called the abscissa.
For the point \, the \-coordinate is \ and for \, it is \.The \- coordinate of a point is its perpendicular distance from the \-axis measured along the \-axis -axis and negative along the negative direction of the \-axis). The \-coordinate is also called the ordinate.For the point \, the \-coordinate is \ and for \, it is \.
Where Is Coordinate Geometry Used In Real Life
Lots and lots of coordinate geometry is used in finding the shortest path and the distance between them. Obviously, there are other concepts involved too. c) It is used in MS Paint to draw curved and slanted lines. d) It is also made in use while developing a game to specify the locations of objects and characters.
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Cartesian Coordinate Planes In Real Life
The Cartesian coordinate plane of x and y works well with many simple situations in real life. For instance, if you are planning where to place different pieces of furniture in a room, you can draw a two-dimensional grid representing the room and use an appropriate unit of measurement. Choose one direction to be x, and the other direction to be y, and define a location as your starting point . You can specify any position in the room with two numbers, in the format , so would be 3 meters in the x-direction and 5 meters in the y-direction, from your chosen point.
You can use this same approach in many situations. All you need to do is define your coordinates, and you can use these to describe locations in the real-world. This is an important part of doing many experiments in physics in particular, or for mapping the locations of populations of organisms in biology. In other settings, your smartphone screen also uses a Cartesian coordinate plane to track where youre touching on the screen, and PDF files or images have a plane to specify locations in the same way.
Main Lesson: Coordinate Graphing Of Real
There are many situations in life that involve two sets of numbers that are related to each other. For example, If you know the price of one ticket for a show, you can calculate the cost for any number of people to attend. Similarly, if you know how much gas will cost for one gallon, you can calculate how many gallons you will be able to purchase with the money in your wallet.
Often, the hardest part of figuring out a complicated problem with lots of data is keeping it all organized so you dont lose track of what you are doing. Using a function table, or T-chart can help you organize information.
A function table has two columns, because it is used to show the relationship between two different strings of numbers. Each function table has a rule, called a function that generates a pattern for one string of numbers when another string of number is used.
The examples below start with one that shows how a function table can be used to figure out how much it would cost for x number of people to attend an afternoon movie that costs $5 per person.
Remind your children that the numbers which are on the same level on the function table are the numbers that are related to each other. Children will sometimes become confused because they try to match numbers which are on different levels on the function table.
So, how does your information get from a function table to a visual display on a coordinate graph? Lets look another example.
2x = y |
= |
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