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Question 29254: Plot the point with coordinates (3, –5).
Answer by sdmmadam@yahoo.com(530)   (Show Source):
You can put this solution on YOUR website! Plot the point with coordinates (3, –5).
Draw two mutually perpendicular straight lines intersecting at O, the origin.
Let the horizontal line be X’OX(the x-axis)
and the vertical line be YOY’(the y-axis)
From the origin go along OX,(that is to the right of O)
a distance OA of 3 units (say cms) and from A on the x -axis
go a distance AP of 5 units below the x-axis
along a line perpendicular to the x-axis.
You arrive at the point P(3,-5)
Note: For a clear picture of the abc of analytical geometry
please go through the following introduction
Let us get introduced to Analytical geometry also called Co-ordinate geometry.
It is the study of geometry using algebra as a tool
Please draw as directed
Draw a horizontal line of sufficient length say 12 to 15 cms.
Mark the right edge as X (capital X) and mark the left edge as X'.
Mark a point O somewhere on the line X'X
This line X'OX we call the x-axis
At O draw a line perpendicular to the x-axis.
Mark the upper edge Y and the lower Y'
This line YOY' is called the y-axis.
Together XOX' and YOY' are called
the rectangular co-ordinate axes of reference.
The point O is called the Origin.
All distances are measured from this point O
While the horizontal line divides the plane of the paper into two parts called the upper half plane and the lower half plane, the veritcal line alone (without the horizontal line) would divide the paper into two parts called the right half plane and the left half plane. Together these two axes divide the plane of the paper into four parts called the quadrants.
The quadrants are numbered in the anti-clockwise direction :
So the portion betweeen OX and OY is called the I quadrant
the portion betweeen OY and OX' is called the II quadrant
the portion betweeen OX' and OY' is called the III quadrant
and the portion betweeen OY' and OX is called the IV quadrant
Any Point in the plane will be situated either on the x-axis, or on the
y-axis,or may be coincident with the origin (which are all special cases
of the location of the point) OR in any one of the quadrants.
Any point in the plane is specified by means of an ordered pair of distances along with their appropriate signs(the horizontal distance first and then the vertical distance next)And this ordered pair of distances along with the appropriate signs are called the co-ordinates of the point P.
How do you decide the signs?
There is what is known as the sign convention:
All distances measured along or parallel to the x-axis from the origin
to the right of the origin are considered positive.
All distances measured along or parallel to the x-axis from the origin
to the left of the origin are considered negative.
All distances measured along or parallel to the y-axis from the origin
above the origin are considered positive.
All distances measured along or parallel to the y-axis from the origin
below the origin are considered negative.
Let us now go a distance of 4 units on some scale(say the cm scale)
from the origin to the right of the origin along the x-axis
and reach the point A.
Then A is a point on the x-axis with length OA = 4 units.
Draw a perpendicular at A to the x-axis. Supposing we go along this perpendicular starting from A, a distance of 7 units (on the same scale)
above the x-axis and arrive at a point B, then we have
AB perpendicular to the x-axis and parallel to the y-axis
with length AB = 7 units.
We say that we have reached the location,
namely the point B in the first quadrant
and this point B is now identified as B=(4,7)
Now if we analyse as to how we reached at the point B,
we know that at first we went a horizontal distance OA=4units
from O along the x-axis and then from A we went above the x-axis
(that is above the origin level) vertically
a distance AB = 7units and arrived at the point B
So B is located in the plane with the help of the horizontal distance
4 units to the right of the origin and a vertial distance 7 units
above the origin
The horizontal distance 4 together with the positive sign
(since we went to the right of the origin) namely +4 forms
what is known as the x- coordinate of the point B in the plane.
The vertical distance 7 together with the positive sign
(since we went above the origin) namely +7 forms
what is known as the y- coordinate of the point B in the plane.
These two co-ordinates
(always the x-coordinate first and then the y-coordinate next
and hence the name ordered pair),separated by a comma put within simple brackets denote the co-ordinates of the point B in the plane.
We write B= (4,7)
If we go a certain distance say 5 units to the left of the origin
from the origin along the x-axis and reach a point A' and from A'
a vertical distance of 9 units above the x-axis,we arrive
at a point C in the II quadrant and C is reached from O by going
to the left a distance of 5units and then by going vertically above,
a distance of 9 units and hence the point C should be C=(-5,9)
If we go a certain distance say 7 units to the left of the origin
from the origin along the x-axis and reach a point B' and from B'
a vertical distance of 8 units below the x-axis,we arrive
at a point D in the III quadrant and D is reached from O by going to the
left a distance of 7units and then by going vertically below, a distance
of 8 units and hence the point D should be D=(-7,-8)
If we go a certain distance say 2 units to the right of the origin
from the origin along the x-axis and reach a point M and from M
a vertical distance of 3.5 units below the x-axis,we arrive
at a point E in the IV quadrant and E is reached from O by going
to the right a distance of 2units and then by going vertically below,
a distance of 3.5 units and hence the point E should be E=(2,-3.5)
What is the practical activity to learn these things
in the correct perspective and never go wrong?
Go to the beach side or a soft sand portion at the back yard.
Mark these axes and the point of intersection O.
Stand at the position O.
Lift one leg up and hop while counting 1 unit for every hop.
Go hopping along OX say a distance of 10(that means 10 hops),
Mark your A or any letter you wish.
Then from there if you hop forwards that is parallel to your OY
that means you are getting into the first quadrant. If you hop backwards
from A(parallel to OY') you are getting into the IV quadrant.
say you have hopped 5 steps along the perpendicular at A forwards and reached the point B.Then B=(10,5).
In the above had you hopped along the same perpendicular but below the x-axis
(which means parallel to OY’) say 2 hops then you have reached a point G in the fourth quadrant and you mark the point G=(10,-2)
Why minus for the y-coordinate? It is because you have gone below the x-axis.
Mark the point and write its co-ordinates as shown above on the sand.
Every time come back to the position O and start your journey afresh.
Stand at the position O.
Lift one leg up and hop while counting 1 unit for every hop.
Go hopping along OX' say a distance of 3
(that means 3 hops),Mark your A' or any letter you wish.
Then from there if you hop forwards that is parallel to your OY
(above the x-axis)that means you are getting into the II quandrant.
If you hop backwards from A'(parallel to OY') you are getting
into the III quadrant.
Let us quickly go through these hops:
Stand at O. Hop along OX(10 hops) Reach A. Hop forwards parallel to OY(6 hops) You have reached your point B in the first quadrant(or any other letter you wish) and B=(10,6)
Stand at O. Hop along OX(7 hops) Reach T. Hop backwards(direction is opposite to the previous vertical distance direction hop.Don't you try to jump backwards and sue me!) parallel to OY'(2 hops) You have reached your point E(or any other letter you wish)in the fourth quadrant and E=(7,-2)
Why minus? Because you have gone below the x-axis. Got it!
Stand at O. Hop along OX'(4 hops) Reach A'. (which means you have come to the left of the origin)
Hop forwards parallel to OY(7 hops) You have reached your point C in the II quadrant(or any other letter you wish) and C=(-4,7)
Why is the first coordinate namely the x-coordinate negative.
Because we have gone to the left of the origin. Right?
Stand at O. Hop along OX'(5 hops) Reach L. Hop backwards(that is below the
x-axis)(direction is opposite to the previous vertical distance direction hop.) parallel to OY'(3 hops) You have reached your point D(or any other letter you wish)in the III quadrant and D=(-5,-3)
Why both the cooddinates negaitive?
You have gone to the left of the origin and hence the x- cordinate is negative and you have gone below the x-axis and hence the y-coordinate is negative.
So now you are tired of hopping.
Relax.Have some juice.
Take a stick or a pointed something (even your pen torch tip) and do these markings on a small scale.
Sit and chant these manthras while jouneying with the stick on the mud
From the Origin
right- positive and above -positive
left- negative and above -positive
left- negative and below- negative
right- positive and below- negative.
Then revise your geography!
From the origin
Go East then positive (x-cordinate positive !)
Go West then negative ((x-cordinate negative !)
Go North then positive (y-cordinate positive !)
Go South then negative ((y-cordinate negative !)
Now on a picnic!
From the origin
Go East and then North.
Where have we arrived?
In the land of the first quadrant.
You are now a point to reside in the I quadrant? What is your identity?
You are always identified by two features namely the x-cordinate
which is always the first identity and then the y-coordinate.
If you have to reside in the I quadrant you have to journey East and then North
and have both your identities positive.
What exactly is your address in the I quadrant?
If you have gone East a distance 4 units,and then North a distance of 3 units
then your house has the address (4,3)
From the origin Go West and then North.
Where have we arrived?
In the land of the II quadrant.
You are now a point to reside in the II quadrant? What is your identity?
You are always identified by two features namely the x-cordinate
which is always the first identity and then the y-coordinate.
If you have to reside in the II quadrant you have to journey West
and then North and have your first identity namely the x-coordinate negative
and the second identity namely the y-coordinate positive.
What exactly is your address in the II quadrant?
If you have gone West a distance 6 units,and then North a distance of 3/2 units then your house has the address (-6,3/2)
From the origin Go West and then South
Where have we arrived?
In the land of the III quadrant.
You are now a point to reside in the III quadrant? What is your identity?
You are always identified by two features namely the x-cordinate which is always the first identity and then the y-coordinate.
If you have to reside in the III quadrant you have to journey West and then South and have both your identities negative
What exactly is your address in the III quadrant?
If you have gone West a distance 4 units,and then South a distance of 3 units
then your house has the address (-4,-3)
From the origin Go East and then South.
Where have we arrived?
In the land of the Fourth quadrant.
You are now a point to reside in the IV quadrant? What is your identity?
You are always identified by two features namely the x-cordinate which is always the first identity and then the y-coordinate.
If you have to reside in the IV quadrant you have to journey East and then South and have your first identity (namely the x-cordinate) positive and the second identity(namely the y-coordinate) negative.
What exactly is your address in the IVquadrant?
If you have gone East a distance 4 units,and then South a distance of 6 units
then your house has the address (4,-6)
In conclusion :So given the co-ordinates of a point we can locate the point.
Supposing with the co-ordinate axes and the origin and a point P on a piece of paper can we determine the co-ordinates of the point P?
YES. We can.
How?
Step 1:Draw a line perpendicular from the given point P to the x-ais
Let it meet the x-axis at M.
Step 2: Now fix your gaze at the origin and follow with your eyes
the track to the point P
Observations: 1. If P is in the I quadrant ,then M is on OX
How to reach P from O
Go OM from O to the right and then go above MP
So the geography is right and then north. so x-coordinate positive
and y-co-ordinate also positive.
Step 3: Measure OM. say 7 units.
Measure MP. say 5 units.
Step 4: Then P=(7,5}
Observations: 2. If P is in the II quadrant ,then M is on OX'
How to reach P from O
Go OM from O to the left and then go above MP
So the geography is left and then north. so x-coordinate negative
and y-co-ordinate positive.
Step 3: Measure OM. say 7 units.
Measure MP. say 5 units.
Step 4: Then P=(-7,5}
Observations: 3. If P is in the III quadrant ,then M is on OX'
How to reach P from O
Go OM from O to the left and then go below MP
So the geography is left and then south. so x-coordinate negative
and y-co-ordinate also negative.
Step 3: Measure OM. say 7 units.
Measure MP. say 5 units.
Step 4: Then P=(-7,-5}
Observations: 4. If P is in the IV quadrant ,then M is on OX
How to reach P from O
Go OM from O to the right and then go below MP
So the geography is right and then south. so x-coordinate positive
and y-co-ordinate negative.
Step 3: Measure OM. say 7 units.
Measure MP. say 5 units.
Step 4: Then P=(7,-5}
What about points on the x-axis?
OKay. Let us mark points A, M, L and T on the x-axis such that
OA = 2 cms, OM = 5 cms, OL = 5 cms, OT = 3.5 cms
with A and M on OX, to the right of the origin and
L and T on OX’ to the left of the origin.
Please mark these points on your paper as directed
What we notice about all these points on the x-axis is that they have
neither gone above the x-axis nor have gone below the x-axis.
This means their y-coordinate = zero
Therefore the points A,M,L and T have their coordinates given as follows:
A=(2,0), M=(5,0)
L=(-5,0) and T=(-3.5,0)
Thus every point on the x-axis has its y-coordinate equal to zero.
we say that the equation to the x-axis is y=0
What about points on the y-axis?
OKay. Let us mark points B, N, G and S on the y-axis such that
OB = 2 cms, ON = 5 cms, OG = 5 cms, OS = 3.5 cms
with B and N on OY, above the origin and
L and T on OY’ below the origin.
Please mark these points on your paper as directed
What we notice about all these points on the y-axis is that they have
neither gone to the right of the origin nor have gone to the left of the origin.
Straight away with out going along the x-axis either to the right or to the left of the origin,they have gone above the origin or below the origin along the y-axis
This means their x-coordinate = zero
Therefore the points B,N,G and S have their coordinates given as follows:
B=(0,2), N=(0,5)
G=(0,-5) and S=(0,-3.5)
Thus every point on the x-axis has its y-coordinate equal to zero.
we say that the equation to the y-axis is x = 0
Now a small piece of activity for your DO IT YOURSELF SESSION
Draw the co-ordinate axes of reference.
Mark the origin.
1)Draw a line parallel to the x-axis at a distance 4 units above the x-axis
Mark 5 or 6 different points say A,B,C,D,E,F on this new parallel line.
(some to the right of the y-axis, some to the left of the y-axis.)
How to find the co-ordinates of these points.
One small clue: Let this parallel line meet the y-axis at J (say)
Give your own measures for JA,JB,JA,JD,JE,JF.
For example JA=2 units
Now give the co-ordinates of the points.
If you get stuck, don’t panic.
Let me give one more clue: You will observe a common feature
for every point on this line parallel to the x-axis and at a distance 4 units above it.
2)Do the above activity for a line parallel to the x-axis at a distance 4 units below it
3)Do the above activity for a line parallel to the y-axis at a distance 4 units to the right of it
4)Do the above activity for a line parallel to the y-axis at a distance 4 units to the left of it
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