Chapter 1: Orienting Yourself: The Use of Coordinates

1.1 Introduction

Coordinate System: A structured framework (like grid lines on a map or graph paper) that enables using numbers to describe exact physical locations of points or objects.
Historical Roots in Bharat:
- Sindhu-Sarasvatī Civilisation: First systematic use of grids for city streets (North–South and East–West directions at uniform distances of about 10 metres apart).
- Baudhāyana (c. 800 C.E.): Used East–West and North–South lines for geometric constructions, developing the Baudhāyana–Pythagoras Theorem and laying the foundation of coordinate geometry.
- Ujjayinī: Described in ancient Siddhāntas (as early as 4th century BCE) as the point marking the central longitude meridian.
- Āryabhaṭa (c. 499 CE): Replaced Greek 'chords' with 'sines', making it easier to calculate coordinates; mapped the sky using Celestial Coordinates from the ecliptic.
- Brahmagupta (c. 628 CE): Formalised the notion and use of zero and negative numbers as algebraic entities, essential for the four-quadrant Cartesian plane.
Later Developments:
- Ptolemy (c. 150 BCE): Described latitudes and longitudes of thousands of locations, including 'Ozine' (Ujjayinī).
- Al-Bīrūnī (c. 1000 CE): Used Indian trigonometric methods to calculate coordinates across Asia; perfected the 'astrolabe'.
- Ömar Khayyām (c. 1100 CE): First mathematician to solve algebraic problems using geometry by interpreting them in terms of coordinates in the plane.
- René Descartes (1637 CE): Formalised that any point in a two-dimensional plane could be defined by two numbers (distances from two perpendicular axes), linking geometry and algebra.
Study in Grades 9 and 10: Students will study this coordinate system, locate objects with pinpoint accuracy, and visualise algebraic equations as geometric shapes.

1.2 Settling In

Scenario: Shalini (Grade 9) uses Coordinate Geometry to guide her visually impaired brother Reiaan in their new city and home.
Method: Shalini used a rectangular grid with pins and threads to represent the room floor, marking key points with pins and connecting object corners with thick wool.
Scale: She used a scale of 1 cm : 1 foot.
Observation: The sketch (Fig. 1.1) shows the floor map; positions of windows cannot be marked on this 2D map.

1.3 The 2-d Cartesian Coordinate System

Definition: A coordinate system using two lines at right angles to each other to mark points in two-dimensional (2-D) space.
x-axis: The horizontal line.
y-axis: The vertical line.
Origin (O): The point of intersection of the x-axis and y-axis, with coordinates (0, 0).
Coordinate Axes: Help locate any point in 2-D space using its 'coordinates'.
Distance Conventions from Origin O:
- Positive: Distances to the right of O (along x-axis) or upwards from O (along y-axis).
- Negative: Distances to the left of O (along x-axis) or downwards from O (along y-axis).
Examples (Fig. 1.2):
- B = (4.5, 0): On the x-axis, 4.5 units to the right of O.
- G = (0, –4.5): On the y-axis, 4.5 units downward from O.
- H = (0, 4): On the y-axis, 4 units above O.
Points on Axes:
- P = (x, 0): Lies on the x-axis. If x is positive, P is right of O; if x is negative, P is left of O.
- P = (0, y): Lies on the y-axis. If y is positive, P is above O; if y is negative, P is below O.
Notation: The ' = ' sign is often dropped, writing P (x, y) instead of P = (x, y).

1.3.1 Exercise Set 1.1

Referring to Fig. 1.3 (Reiaan’s room), answer the following questions:

(i) Door D1R1 position: How far is the door from the left wall (y-axis) and the x-axis?
(ii) Coordinates of D1: What are they?
(iii) Door width (R1=(11.5, 0)): How wide is the door? Is this a comfortable width for the room door? Accessible for a person in a wheelchair?
(iv) Bathroom door (B1(0, 1.5), B2(0, 4)): Is the bathroom door narrower or wider than the room door?

1.3.2 Think and Reflect (Post Exercise Set 1.1)

1. Standard door widths: What are the standard widths for a room door?
2. School door accessibility: Are the doors in your school suitable for people in wheelchairs?

Cartesian Plane: The plane in which the axes are situated; also called the coordinate plane or the xy-plane.
Quadrants: The axes divide the plane into four parts, numbered as shown in Fig. 1.4 (Quadrant I: top-right, Quadrant II: top-left, etc.).
Coordinate Signs in Quadrants:
- Quadrant I: Both x- and y-coordinates are positive (+, +).
- Quadrant II: Negative x-coordinate and positive y-coordinate (–, +).
- Quadrant III: Both x- and y-coordinates are negative (–, –).
- Quadrant IV: Positive x-coordinate and negative y-coordinate (+, –).
Meaning of Coordinates P(x, y):
- x: Represents the perpendicular distance of P from the y-axis, measured along the x-axis (x-coordinate).
- y: Represents the perpendicular distance of P from the x-axis, measured along the y-axis (y-coordinate).
Examples (Fig. 1.4):
- S (3, –5): In Quadrant IV, x-coordinate is 3, y-coordinate is –5.
- Q (–5, 3): In Quadrant II, x-coordinate is –5, y-coordinate is 3.

1.3.3 Think and Reflect (Post Quadrants)

1. x-coordinate on y-axis: What is the x-coordinate of a point on the y-axis?
2. y-coordinate on x-axis: Is there a similar generalisation for a point on the x-axis?
3. Coincidence of Q(y, x) and P(x, y): Does point Q (y, x) ever coincide with point P (x, y)? Justify.
4. Claim: If x ≠ y, then (x, y) ≠ (y, x); and (x, y) = (y, x) if and only if x = y: Is this claim true?

1.3.4 Exercise Set 1.2

On a graph sheet, mark the x-axis and y-axis and the origin O. Mark points from (–7, 0) to (13, 0) on the x-axis and from (0, –15) to (0, 12) on the y-axis (Use the scale 1 cm = 1 unit). Using Fig. 1.5, answer the given questions:

1. Reiaan’s rectangular study table: Place with three feet at (8, 9), (11, 9) and (11, 7).
- (i) Where will the fourth foot of the table be?
- (ii) Is this a good spot for the table?
- (iii) What is the width of the table? The length? Can you make out the height of the table?
2. Bathroom door: If the bathroom door has a hinge at B1 and opens into the bedroom, will it hit the wardrobe? Are there any changes you would suggest if the door is made wider?
3. Reiaan’s bathroom:
- (i) What are the coordinates of the four corners O, F, R, and P of the bathroom?
- (ii) What is the shape of the showering area SHWR? Write the coordinates of its four corners.
- (iii) Mark off a 3 ft × 2 ft space for the washbasin and a 2 ft × 3 ft space for the toilet. Write the coordinates of the corners of these spaces.
4. Other rooms in the house:
- (i) Reiaan’s room door leads from the dining room (length 18 ft, width 15 ft). Length extends from P to A. Sketch the dining room and mark the coordinates of its corners.
- (ii) Place a rectangular 5 ft × 3 ft dining table precisely in the centre of the dining room. Write down the coordinates of the feet of the table.

1.4 Distance Between Two Points in the 2-D Plane

Known Distances: We can find the distance between two points if they are on the axes or form a line segment parallel to the axes.
General Distance: For segments not parallel to either axis, the Baudhāyana–Pythagoras theorem is used to find the distance between any two points in the xy-plane.

1.4.1 Derivation Example (Triangle ADM, Fig. 1.6 & 1.7)

Points: A (3, 4), D (7, 1).
Distance along x-axis (CD): x-coordinate of D – x-coordinate of A = 7 – 3 = 4 units.
Distance along y-axis (AC): y-coordinate of A – y-coordinate of D = 4 – 1 = 3 units.
Distance AD: Using Baudhāyana–Pythagoras Theorem: AD = √(4² + 3²) = √(16 + 9) = √25 = 5 units.
Similarly for DM and MA:
- DM: √((9-7)² + (6-1)²) = √(2² + 5²) = √(4 + 25) = √29 units.
- MA: √((9-3)² + (6-4)²) = √(6² + 2²) = √(36 + 4) = √40 units.

1.4.2 General Distance Formula

Formula: The distance between points (x₁, y₁) and (x₂, y₂) is given by √((x₂ - x₁)² + (y₂ - y₁)²).
Sign Independence: It makes no difference whether (x₂ - x₁) and (y₂ - y₁) are positive or negative due to squaring.
Negative Coordinates: The formula is valid even if x₁, x₂, y₁, y₂ take negative values.

1.4.2.1 Reflection Example (Fig. 1.9)

Triangle AMD reflected in the y-axis, creating A'M'D'.

Original Points: A (3, 4), D (7, 1), M (9, 6).
Reflected Points: A' (–3, 4), D' (–7, 1), M' (–9, 6).
C'D' (x-distance for A'D'): x-coordinate of A' – x-coordinate of D' = –3 – (–7) = 4.
A'C' (y-distance for A'D'): y-coordinate of A' – y-coordinate of D' = 4 – 1 = 3.
Distance A'D': Using Baudhāyana–Pythagoras Theorem: √(4² + 3²) = 5 units.
Distances D'M' and M'A':
- D'M': √((–9 – (–7))² + (6 – 1)²) = √((–2)² + 5²) = √(4 + 25) = √29 units.
- M'A': √((–9 – (–3))² + (6 – 4)²) = √((–6)² + 2²) = √(36 + 4) = √40 units.
Observation: Reflection preserves the lengths of the sides of the triangles.

1.4.3 Think and Reflect (Post Reflection)

1. Changes in reflection: What has remained the same (e.g., lengths, shape) and what has changed (e.g., coordinates, quadrant)?
2. Reflection in x-axis: Would these observations be the same if ΔADM is reflected in the x-axis (instead of the y-axis)?

End-of-Chapter Exercises

1. Origin Coordinates: What are the x-coordinate and y-coordinate of the point of intersection of the two axes?
2. Point W & H: Point W has x-coordinate equal to –5. Can you predict the coordinates of point H which is on the line through W parallel to the y-axis? Which quadrants can H lie in?
3. Points R(3, 0), A(0, –2), M(–5, –2), P(–5, 2): If they are joined in the same order, predict:
- (i) Two sides of RAMP that are perpendicular to each other.
- (ii) One side of RAMP that is parallel to one of the axes.
- (iii) Two points that are mirror images of each other in one axis. Which axis will this be?
4. Plot Z(5, –6): Plot point Z(5, –6) on the Cartesian plane. Construct a right-angled triangle IZN and find the lengths of the three sides. (Answers may differ).
5. Coordinate system without negative numbers: What would a system of coordinates be like if we did not have negative numbers? Would this system allow us to locate all the points on a 2-D plane?
6. Collinearity check (M(–3, –4), A(0, 0), G(6, 8)): Are the points M(–3, –4), A(0, 0) and G(6, 8) on the same straight line? Suggest a method to check this without plotting and joining the points.
7. Collinearity check (R(–5, –1), B(–2, –5), C(4, –12)): Use your method (from Problem 6) to check if the points R(–5, –1), B(–2, –5) and C(4, –12) are on the same straight line. Now plot both sets of points and check your answers.
8. Plot vertices using origin as one vertex:
- (i) A right-angled isosceles triangle.
- (ii) An isosceles triangle with one vertex in Quadrant III and the other in Quadrant IV.
9. Midpoint of ST (Table): Complete the table below regarding whether M is the midpoint of segment ST and the reason.
S M T Is M the midpoint of ST? Yes or No Reason for your answer
(–3, 0) (0, 0) (3, 0)
(2, 3) (3, 4) (4, 5)
(0, 0) (0, 5) (0, –10)
(–8, 7) (0, –2) (6, –3)
When M is the mid-point of ST, can you find any connection between the coordinates of M, S and T?
10. Find B coordinates: Use the connection you found to find the coordinates of B given that M (–7, 1) is the midpoint of A (3, –4) and B (x, y).
11. Trisection points P, Q of AB: Let P, Q be points of trisection of AB, with P closer to A, and Q closer to B. Using your knowledge of how to find the coordinates of the midpoint of a segment, how would you find the coordinates of P and Q? Do this for the case when the points are A (4, 7) and B (16, –2).
12. Points on a circle K (center O(0,0)):
- (i) Given the points A (1, –8), B (–4, 7) and C (–7, –4), show that they lie on a circle K whose center is the origin O (0, 0). What is the radius of circle K?
- (ii) Given the points D (–5, 6) and E (0, 9), check whether D and E lie within the circle, on the circle, or outside the circle K.
13. Midpoints of triangle ABC sides: The midpoints of the sides of triangle ABC are the points D, E, and F. Given that the coordinates of D, E, and F are (5, 1), (6, 5), and (0, 3), respectively, find the coordinates of A, B and C.
14. City Model (2 main roads, 10 streets each direction, 200m apart): A city has two main roads which cross each other at the centre of the city (N–S and E–W directions). All other streets run parallel to these roads and are 200 m apart. There are 10 streets in each direction.
- (i) Using 1 cm = 200 m, draw a model of the city in your notebook. Represent the roads/streets by single lines.
- (ii) Each street intersection is formed by two streets (N–S and E–W). If the second street running N–S and 5th street in E–W meet, it's (2, 5). Using this convention, find:
  - (a) how many street intersections can be referred to as (4, 3).
  - (b) how many street intersections can be referred to as (3, 4).
15. Computer Graphics (Screen 800x600 pixels, origin bottom-left): A computer graphics program displays images on a rectangular screen. A circular icon of radius 80 pixels is drawn with its centre at A (100, 150). Another circular icon of radius 100 pixels is drawn with its centre at B (250, 230). Determine:
- (i) whether any part of either circle lies outside the screen.
- (ii) whether the two circles intersect each other.
16. Plot A(2, 1), B(–1, 2), C(–2, –1), D(1, –2): Plot these points in the coordinate plane. Is ABCD a square? Can you explain why? What is the area of this square?

S	M	T
(–3, 0)	(0, 0)	(3, 0)
(2, 3)	(3, 4)	(4, 5)
(0, 0)	(0, 5)	(0, –10)
(–8, 7)	(0, –2)	(6, –3)

Chapter Summary

Locating Position: To locate an object or point in a plane, two perpendicular lines are required – one horizontal (x-axis) and one vertical (y-axis).
Plane Names: The plane is called the Cartesian plane, the coordinate plane, or the xy-plane; the lines are called the coordinate axes.
Axes Orientation: The horizontal line is the x-axis, and the vertical line is the y-axis.
Quadrants: The coordinate axes divide the plane into four parts called quadrants.
Origin: The point of intersection of the axes is called the origin, with coordinates (0, 0).
Coordinates of a Point (x, y):
- x-coordinate (abscissa): The distance of a point from the y-axis.
- y-coordinate (ordinate): The distance of a point from the x-axis.
Points on Axes:
- The coordinates of a point on the x-axis are of the form (x, 0).
- The coordinates of a point on the y-axis are of the form (0, y).
Coordinate Signs by Quadrant:
- Quadrant I: (+, +)
- Quadrant II: (–, +)
- Quadrant III: (–, –)
- Quadrant IV: (+, –)
Equality of Coordinates: If x = y, then (x, y) = (y, x). If x ≠ y, then (x, y) ≠ (y, x).
Distance between Points:
- Horizontal line (x₁, y) and (x₂, y): The absolute value |x₂ – x₁|.
- Vertical line (x, y₁) and (x, y₂): The absolute value |y₂ – y₁|.
- General points (x₁, y₁) and (x₂, y₂): By the Baudhāyana–Pythagoras Theorem, the distance is √((x₂ - x₁)² + (y₂ - y₁)²).