Orienting Yourself : The Use of Coordinates[ Coordinates का use करके अपनी दिशा या स्थान जानना ]
1.1 Introduction 
A system of coordinates is a structured framework ( like the grid lines on a map or graph paper ) that enables us to use numbers to describe the exact physical locations of points or objects.
The idea of ' grid - based thinking ' and the geometry required to define the locations of points in space - indeed has deep roots in Bharat.
Explanation : Coordinate system ek aisa organised framework hota hai jisme hum numbers ki help se kisi bhi point ya object ki exact location bata sakte hain. Yeh bilkul map ya graph paper ki grid lines ki tarah hota hai. Graph paper par jo horizontal aur vertical lines hoti hain, woh milkar ek grid banati hain, aur usi grid ki help se hum kisi point ki sahi position find karte hain.
Jaise agar kisi point ki location ( 3, 2 ) ho, to iska matlab hota hai 3 units right aur 2 units upar jana. Is tarah coordinates kisi jagah ki exact position clear kar dete hain.
“grid-based thinking” ka matlab hai jagahon ko lines aur boxes ke form me sochna aur geometry ki help se unki location define karna. Geometry hume distance, direction aur points ki position samajhne me help karti hai. Fir paragraph yeh bhi batata hai ki aisi thinking aur geometry ka concept sirf modern time ka nahi hai, balki Bharat me bhi bahut purane samay se iska use hota aa raha hai, jaise land measurement, city planning, temple construction aur astronomy me.
1. Sindhu - Sarasvati Civilisation ( 1000 years ) : The first systematic use of grids occured thousands of years ago - on massive urban scale( “Bahut bade shahri star par”),where city streets were constructed with striking precision(accurate) in North - South and East - West directions at uniform distances of about 10 metres apart.
2. Baudhayana ( 800 C.E ) : As we have seen ,later used East - West and North - South lines for his deep geometric constructions, developing the Baudhayana Pythagoras Theorem and thus laying the foundation of coordinate geometry. Putting coordinates on the earth's surface later became important for navigation.
Explanation : East–West aur North–South lines → direction wali seedhi lines.
Geometric constructions → geometry ke shapes aur measurements banana/samajhna.
Baudhayana Pythagoras Theorem → Pythagoras theorem jaisa important rule,or the distance between two points.
Foundation of coordinate geometry → coordinate geometry ki shuruaati neev dali.
Earth ki surface par coordinates ka use navigation (raasta dhoondhne aur travel karne) ke liye bahut important ho gaya.
Ujjain ko prachin samay me, kam se kam 4th century BCE se, purane Siddhantas me ek special point maana gaya tha.
Ye point central longitude meridian ko mark karta tha, jiske basis par doosri jagahon ki position measure ki jaati thi.
3. Greek mathematician Ptolemy ( 150 BCE ) : Ptolemy ne purane mathematicians( Hipparchus ) ke ideas ka use karke duniya ki bahut si jagahon ke coordinates ( longitude and latitude ) bataye jisme ' Ozone ' bhi saamil tha.
4. Aryabhata ( 499 CE ) : Greek Chords → purana mathematical method ko replace Kiya.
Sines → trigonometry ka easier method.
Coordinates of a star or city → star ya city ki exact position.
Celestial coordinates → sky me stars/planets ki positions batane wali coordinate system.
Ecliptic → sky me Sun ke movement ka imaginary path
5. Brahmagupta ( 628 CE ) : formalised the notion and use of zero and the negative numbers as algebraic entities. Brahmagupta 's work was translated into Arabic and the Ujjayini meridian entered Arabic geography under the name ' Arin ' serving as the zero - longitude reference for early Arabic mappswhich also then made use of negative numbers.
Explanation : “Brahmagupta ne zero aur negative numbers ke idea ko properly rules ke saath define kiya aur unhe algebra me numbers ki tarah use karna shuru kiya.”
Brahmagupta ke kaam ko Arabic language me translate kiya gaya.
Ujjain ki meridian line Arabic geography me “Arin” naam se famous hui aur ise early Arabic maps me zero-longitude reference point ki tarah use kiya gaya.
In maps me negative numbers ka bhi use hone laga.
6. Al - Biruni ( 1000 CE ) : The influential(prabhaavsaali) Arab scholar Al-Bīrūnī (c. 1000 CE) travelled to India, studied the Siddhāntas, and used Indian trigonometric methods to calculate the coordinates of various cities across Asia.Al-Bīrūnī also later perfected the ‘astrolabe’, a handheld device that allowed sailors to find their coordinates by looking at the stars.
Explanation : prabhaavsaali arbi vidvaan Al - Biruni lagbhag 1000 CE me bharat aae .usne siddhantao ka study kiya or indian trigonometry process ka use karke asia ke kai sahro ki position ki ganna ki, baad me usne “astrolabe'yantra ko behtar banaya। Yah ek haath me pakra jaane vala yantra tha jisse naavik star ko dekhkar apni position ka pata laga lete.
7. Omar Khayyam ( 1100 CE ) : Omar Khayyām (c. 1100 CE), who had become an expert in the Indian decimal system and algebraic formalism, was the first mathematician to solve algebraic problems using geometry by interpreting them in terms of coordinates in the plane.
Explanation :
Omar Khayyam ,jo Bhartiya decimal system or algebraic formalism ke expert ban gae the, vah pahle mathematician the jinhone geometry ki help se algebra ke questions ko solved kiye or unhone plane me coordinate ka help lekar un question ko samjha or solve Kiya.
Equations ko shapes aur coordinates se samjha.
8. Fermat and Descartes ( 1636 - 1637 CE ) : 12th century me ye knowledge Europe pahunchi. René Descartes ne bataya, Har point ko 2 numbers se show kar sakte hain.
Example: (3,2)
3 = horizontal distance
2 = vertical distance
Isi ko Cartesian coordinate system kehte hain.
Ab:
equations ko graph me draw kar sakte the
shapes ko equations me likh sakte the
Isi se modern coordinate geometry bani.
1.2 Settling In ( नये जगह में सेट होना )
Shalini ne Reiaan ko naye ghar aur room ka layout samjhane ke liye ek sketch/map banaya. Yeh map room ke floor ka top view hai, matlab upar se dekhne par room kaisa dikhega wahi show kiya gaya hai. Usne coordinate geometry ka idea use kiya taaki Reiaan easily room ki positions samajh sake.
Sketch me important points ko pins se mark kiya gaya. Jaise room ke corners, bathroom ke corners aur furniture ke corners. Shalini ne ek scale use kiya: 1 cm = 1 foot.
Matlab real life me agar koi object 12 foot ka hai, to map me usko 12 cm se show kiya jayega.
Objects ke corners ko thick wool/thread se connect kiya gaya taaki Reiaan apni fingers se touch karke objects ki shape aur position feel kar sake. Isse usko samajhne me aasani ho.
Figure me left side par bathroom dikhaya gaya hai jiska size 6 ft × 9 ft hai. Bathroom ke andar bathing area bhi show kiya gaya hai. Right side par bedroom hai jiska size 12 ft × 10 ft hai. Bedroom me bed aur wardrobe rakha hua hai. Wardrobe ka size 4 ft × 2 ft diya gaya hai.
Figure me doors bhi bane hue hain. Door ke paas curved dotted lines dikh rahi hain jo batati hain ki door kis direction me open hota hai.
Yeh figure mainly yeh samjhana chahta hai ki coordinate geometry aur maps ki help se kisi bhi jagah ya object ki exact position batayi ja sakti hai. Yeh real life me room maps, city maps aur navigation me use hota hai.
Do you see why the position of the windows cannot be marked on this map ?
Ans : The position of the windows cannot be marked on this map because a map shows only a top view of the place, while windows are on the walls and involve height, which cannot be shown clearly on a flat map.
1 .3 The 2 - D Cartesian Coordinate System
This graph is a Cartesian Coordinate System.
There are two lines:
Horizontal line = x-axis
Vertical line = y-axis
Where both lines meet is called the Origin. 
Origin = O (0,0)
Positive and negative directions:
Right side of x-axis = positive (+)
Left side of x-axis = negative (−)
Upward on y-axis = positive (+)
Downward on y-axis = negative (−)
Points on the Graph
1. Point O = (0,0)
This is the origin.
x = 0
y = 0
It means the point is exactly at the center where both axes meet.
2. Point B = (4.5, 0)
Point B lies on the x-axis.
First number 4.5 = movement on x-axis
Second number 0 = no movement on y-axis
So:
Move 4.5 units to the right
Move 0 units up/down
That is why B is on the x-axis.
3. Point E = (−2.9, 0)
Point E is also on the x-axis.
x = −2.9
y = 0
Negative x means move to the left.
So:
Move 2.9 units left
No vertical movement
Therefore E is left side of origin on x-axis.
4. Point H = (0, 4)
Point H lies on the y-axis.
x = 0 → no left/right movement
y = 4 → move upward
So:
Stay on y-axis
Move 4 units up
That is why H is above the origin.
5. Point G = (0, −4.5)
Point G is also on the y-axis.
x = 0
y = −4.5
Negative y means downward movement.
So:
No left/right movement
Move 4.5 units downward
That is why G is below the origin.
Important Rule of Coordinates
A point is written as:
First number = x-coordinate
Second number = y-coordinate
Example:
(4, 2) → first go 4 right, then 2 up
(−3, 1) → first go 3 left, then 1 up
While writing the coordinates of a point,it is often convenient to drop the ' = ' sign and write P = ( x ,y ) simply as P ( x, y) . This is specially true while marking points on a graph.
It means,P = ( x , y ) nahi graph par hm ise P ( x, y ) likh sakte hai.
Exercise Set 1.1
Figure. 1.3 shows Reiaan's room with points OABC marking it's corners. The x - and y - axes are marked in the figure. Point O is the origin.
Referring to Fig.1.3 answer the following questions :
(i) If D_1 R_1 represents the door to Reiaan's room,how far is the door from the left wall ( the y - axis ) of the room ? How far is the door from the x -axis ?
(ii) What are the coordinates of D_1 ?
(iii) If R_1 is the point ( 11.5, 0 ) ,how wide is the door ? Do you think this is a comfortable width for the room door ? If a person in a wheelchair wants to enter the room, will be/she be able to do so easily ?
(iv) If B_1 (0, 1.5 ) and B_2 (0, 4 ) represent the ends of the bathroom door, is the bathroom door narrower or wider than the room door ?
Solutions : From the figure,
The room door is represented by points D_1 and R_1.
R_1 = ( 11.5, 0 )
D_1 is at x = 8 on the x -axis,so D_1 = ( 8, 0 )
( i) Distance of the door from the left wall and x - axis :
The left wall is the y - axis
Since D_1 is at x = 8,
Distance from the y - axis = 8 units.
The door lies on the x - axis itself,so
Distance from the x - axis = 0 units.
Answer : From left wall ( y - axis ) = 8 units.
From x - axis = 0 units.
(ii) Coordinates of D_1
From the graph,
D_1 = ( 8, 0 )
Answer : ( 8, 0 )
(iii) Width of the door
Given,
R_1 = ( 11,5, 0 )
D_1 = ( 8 , 0 )
Width of the door = 11.5 - 8 = 3.5
Door width = 3.5 units .
This is quite a wide door,so it would be comfortable. A wheelchair user should also be able to enter easily because the door is wide enough.
Answer :
- Width = 3.5 units
- Yes, it is comfortable .
- Yes,a wheelchair can enter easily .
(iv) Width of bathroom door
Bathroom door endpoints :
B_1 = ( 0, 1.5 ) , B_2 = ( 0, 4 )
Width = 4 - 1.5 = 2.5 units
Now,
Bathroom door width = 2.5 units
Room door width = 3.5 units
Since,
2.5 < 3.5
Answer : The bathroom door is narrower than the room door.
