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Coordinate geometry


In coordinate geometry we will be working on a plane, a 2d plane.



This is how a coordinate plane looks. A coordinate plane is just a 2d sheet with values.

There are two evident lines in this plane. One vertical line and one horizontal line and both of these are going through each other at point (0,0). These are two axes of a coordinate plane.

Similar to a graph, the horizontal line is the x-axis, and the vertical line is the y-axis.

A graph sheet has 1/4th of a coordinate plane.



In a graph, you only see the shaded part in this image.


The positive side of the x-axis is denoted by x, and the negative side of the x-axis is denoted by X’ [dash]. Similarly, the positive side of the y axis is denoted by Y and the negative side by Y’ [dash].


You can also denote Y as north, X as East, Y dash as south, and X dash as west.

A coordinate system is used to put the world on a 2d plane, which aids us in calculating various pieces of information.


Coordinate geometry has various applications in real life. It is used in air traffic, GPS, map projection, and in assigning latitude and longitude.


A coordinate plane on its own is of no use. We use the coordinate plane to mark the location by specific coordinates.


A coordinate is a pair of x and y coordinates. An example of a coordinate is (2,3). The first value denotes the x-coordinate, and the second value is the y-coordinate. The x-coordinate tells us the horizontal distance of a point from the origin, which is the point (0,0) The y-coordinate tells us the vertical distance of a point from the origin.


Imagine you want to go from location A to location C. First, you go 2km east to reach location B. Then you travel 2km north to reach location c. This is the path you take to reach point c based on the path available.

Now imagine these three points or an empty field, with no specific path. Would you still take this route?






Anyone would directly go from A to C skipping B, as it would be the shortest path. The shortest path would be:


The shortest path would be the line denoted by h. It is our instinct to always follow this path, as we feel it is the shortest.










The shortest path, h has something called a slope. Slope is the steepness of a line.

The slope of a line is measured by the formula:

The slope of a line is basically the how much you rise divided by how much you walk. In the diagram above, we have a triangle that can help us. The rise here would be two, because the change in the y-coordinates from A-C is two. The walk would also be two, because the change in the x-coordinates from A-C is two.

This will give us a slope of 2/2=1. The slope of this line h is 1, as it increments by 1 unit in the plane.


This same formula can be altered to give a much easier to use formula. That is:




When using this formula, we don't have to draw a walk and a rise line. All we need are the coordinates of the two points, A and C. The coordinates of A are (1,1) and the coordinates of C(3,3). First we need to denote one pair as x1 and y1 and another pair as x2 and y2. Which pair is donated as which doesn't matter.

Suppose, coordinates of A are x1y1, then coordinates of C will be x2y2. We need to substitute this in the formula and solve it.

This will give us a slope of 1. We got the same value using the first formula, so both of them are the same. They are the same because in the first formula the numerator is the rise and in the second formula the numerator is the change in y-coordinate and y-coordinate is nothing but the rise. On the other hand, walk is the same as change in x-coordinates.

Coordinates

Solution

(2,3) and (3,4)

1

(5,3) and (8,2)

​-1/3

​(-3,2) and (2,2)

0

(6,2) and (6,5)

undefined

In this table given, the last two values are weird and unusual because one of them is a straight line and the other one is a horizontal line.

The line formed by connecting (-3,2) and (2,2) would be:



This will produce a straight line, because both the coordinates have the same y-coordinate, which means both have risen to the same value giving a horizontal line. The slope of a horizontal line is always zero, because there is no steepness in this line. It is straight line.


One the other hand (6,2) and (6,5) gave the slope as undefined. A line has an undefined slope when it is a vertical line. Vertical lines have an infinitely large slope which is denoted by undefined. In mathematical terms, these two coordinates have the same x-coordinate which means there is a change only in the y-coordinate. When there is no change in the horizontal distance, it produces a straight line. The walk value of a straight line is zero and anything divided by zero is undefined.


Distance formula:


We will move on to finding the distance between two plotted coordinates. In a coordinate system, the units denote a specific value. For example, 1 box in the plane can denote 1km or 1m or 1mile.

Finding the distance is self-explanatory. You are finding the distance between two locations on a coordinate plane.

A real-life application of this topic is in maps. All world maps are coordinate systems.


In google maps, when you choose a specific location, it gives you the coordinates of that location. It is based on this scale.


Now we find out exactly distance is calculated on a coordinate plane.















In this image different buildings are plotted on a coordinate plane with scale of 1box=1km.

When we are finding the distance we are finding the distance of the shortest path between the two coordinates.


The distance formula is:




Distance between bank and park:

First you need to write down the coordinates of the two locations and choose x1y1 and x2y2:


bank=(2,3) as x1 and y1

park=(4,2) as x2 and y2


After this you have to substitute the values and solve the equation.

The distance between the bank and park is 2.24km.










Mid-point formula:


Mid point formula is used to the find the exact middle of two points.













It is basically the average of the two x coordinates put together with the average of the two y coordinates.

The mid point formula is:




With this we will find the mid point of park and toy-store.

coordinates of park: (4,2) as x1y1

coordinates of toy-store: (8,5) as x2y

The coordinates of the mid point of park and toy store are (6,3.5).








X and Y intercepts:


X-intercept - the point at which the line passes through the X axis.

Y-intercept - the point at which the line passes through the Y axis.


The line need not have both the x and y intercept. A line can either one of these two, but a line will never not have both of these.


The first image is of a line with only a x-intercept. The second image is of a line with both the x and y intercepts. The third image is of a line with only a y-intercept.


We will use this information in the next section of equation of line.

Equation of line:


Equation of line is basically an equation that when plotted on the coordinate plane will produce a specific line. Equation of a line is basically a mathematical way to represent a line.


There are three methods that can be used to find the equation of line:

Point slope form:



X1 and y1 are coordinates of any one point in the line. X and Y will remain as x and y, you won’t substitute them for anything. M here again represents the slope.

So, to use this formula you need to know one set of coordinates and the slope of the line.

Slope intercept form:



Over here y and x represent coordinates of any one point in the line. M represents slope and c represents in the y intercept. So, we can only use this method if we have two of these three information given here. The final equation of line won't specify any x and y value. all x and y values on the line can be used.

Double intercept form:




A represents the x intercept, B represents the y intercept and X and Y here you will leave it as it is.


Since there are so many methods to find the equation of a line which will give the equation of line in different forms, we also have a standard way to write equation of line and that is:


Ax+By=C

A, B and C have to be integers. A has to be a positive integer.


What do you think will be the equation of the y-axis?

It would be x=0. Throughout the y-axis the value of x remains zero, as the horizontal distance doesn't change throughout this line. x=0 will always produce the y axis, as it doesn't stand true to any other line.


Similarly, what do you think will be the equation of x-axis?

In this case, it would be y=0. Through the x-axis, only the horizontal distance from origin is changing, the vertical distance from origin remains zero, which gives the equation of line, y=0.


Example1:

The two coordinates we have here are actually x and y intercepts.

First, we need to calculate the slope, using the two points (0,46) and (1820,0).


Slope:

(0,46) as x1y1

(1820,0) as x2y2









Equation of line:


Double intercept form:

When finding the equation of line you have to choose the best formula to use among the three. The best one to use here would be the double intercept form as we know both the x and y intercepts. We can use the other two methods too.

This is the equation of this line in double intercept form. We already know the x and y intercept, so all we have to do is substitute it in the equation.



Converting from double intercept to standard form:

This equation is the same as the line that is in the diagram. You can go to any online coordinate plotters and type in this equation. The resulting line would be the same line that is in the diagram.







slope intercept form:

An equation in slope intercept form looks like: y=4x+5

Over here there isn't any calculations to do. The two values that needs to be there in the equation are the slope and y-intercept. We already know these values, so we just have to substitute them in.


slope intercept form to standard form:













Point slope form:

(0,46) as x1y1.

We already have all the information. We have to choose either on of the points take their coordinates substitute it in the equation and substitute in the slope value we got.


Point slope to standard form:





















Example2:

Over here we will find the equation of line of the line BC.

Slope:

coordinates of B(-25,-20) as x1y1

coordinates of C(-10,10) as x2y2











Slope intercept form:

You can use any coordinates of any point on the line to find the value of c (y-intercept).

(-25,-20) as x and y.

You might say the line in the diagram doesn't have a y-intercept. It doesn't have a y-intercept because it is a segment and not a line. When we are using equation of line, we are taking the slope and coordinates of a segment to construct a line. This constructed line can have a y-intercept.

A segment is a part of a line that has two endpoints.

All the lines in the diagram are segments and not lines. Lines are infinitely long and extend in both directions. The segments in the diagram are a small part of a line.



Converting from slope intercept to standard form:








Point slope form:

(-10,10) as x1y1






Point slope to standard form:












Double intercept form:


Finding y-intercept:

(-10,10) as x1y1


















equation of line:







Double intercept to standard form:










Relationship between perpendicular lines:


When you have two perpendicular lines, the product of their slopes will be -1.


Example:

Question:

Another ladder is placed at the center of this ladder in such a way that it reaches the window of the adjacent building. Both the ladders are perpendicular to each other. What is the slope of the second ladder?


Slope of second ladder:

We don't have points on the second ladder to calculate the slope, so we will use the perpendicular rule.

The slope of the first ladder is -1.6.

The product of both these slopes should be -1, so:















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