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Use of Coordinates Class 9 PPT & Notes – Ganita Manjari Ch 1

Class IX Maths 🖥️ PPT Slides Ch 1: Orienting Yourself: The Use of Coordinates NCERT book: Ganita Manjari

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Orienting Yourself: The Use of CoordinatesClass 9 Maths · Chapter 1 · Ganita Manjari Part I (2026-27) · Notes with diagrams and practice questions

9sections

22MCQs

14Q&A

11diagrams

18PPT slides

Complete chapter notes written from our 18-slide PPT, with the exercise answers worked out and checked, followed by practice questions. Tap an option in the MCQ section to check it at once, and tap Check answer anywhere for the written answer. To get the full 18-slide PPT, use the Download PDF button at the top of this page.

Section 1 · Chapter overview

What this chapter covers

What is a coordinate system?
History: India’s contribution to coordinates
The 2-D Cartesian plane: x-axis, y-axis and origin
The four quadrants: I, II, III and IV
Plotting points on the plane
The distance between two points, and the Baudhayana–Pythagoras theorem in coordinates

India’s contribution to coordinates

Who or what	When	Contribution
Sindhu-Sarasvati Civilisation	3000+ BCE	Planned cities with streets running north–south and east–west; to find a shop you count along the streets from the centre
Ujjayini	Ancient India	Used as the “zero meridian” of ancient Indian astronomy, like India’s Greenwich
Baudhayana	c. 800 BCE	Used east–west and north–south lines for geometry and proved the Baudhayana–Pythagoras theorem, the backbone of the distance formula
Aryabhata	c. 499 CE	Mapped the sky using coordinates; replaced chords with sines, which made calculating star positions easier
Brahmagupta	c. 628 CE	Formalised zero and negative numbers, without which the four-quadrant plane would be impossible
René Descartes	1637 CE	Formalised 2-D coordinates, which is why the system is called Cartesian

Mapping Reiaan’s room

Reiaan maps the family’s room on a coordinate grid, where each pin is a point (x, y) and the threads show the walls and furniture. The bedroom is 12 ft × 10 ft, the bathroom is 6 ft × 9 ft, and the wardrobe is 4 ft × 2 ft. The scale is 1 cm = 1 foot, and the corner of the room is the origin O (0, 0).

Think: why can’t we show windows on this floor map? Because windows are on the walls, not on the floor.

A floor plan of Reiaan’s bathroom and bedroom drawn on a coordinate grid from 0 to 12 feet — Slide 3: mapping Reiaan’s room

Section 2 · The Cartesian plane

The 2-D Cartesian coordinate system

Two perpendicular number lines are all we need
The x-axis is the horizontal line. To the right of 0, x is positive (+); to the left of 0, x is negative (−)
The y-axis is the vertical line. Above 0, y is positive (+); below 0, y is negative (−)
The origin O = (0, 0) is where the axes intersect, the starting point of all measurements
x comes first, then y: always write (x, y)

Point	Where it lies
H = (0, 4)	On the y-axis, positive y, above O
B = (4.5, 0)	On the x-axis, positive x, to the right of O
E = (−2.9, 0)	On the x-axis, negative x, to the left of O
G = (0, −4.5)	On the y-axis, negative y, below O

The Cartesian plane with the x-axis, y-axis and origin, and the points H, B, E and G marked on the axes — Slide 4: the 2-D Cartesian coordinate system

The four quadrants

Quadrant	Signs of (x, y)	Example
I	(+, +): both positive	P (5, 4)
II	(−, +): x negative, y positive	Q (−5, 3)
III	(−, −): both negative	R (−3, −4)
IV	(+, −): x positive, y negative	S (3, −5)

Quick memory trick

I = both +. II = x −, y + (left but up). III = both − (deep underground). IV = x +, y − (right but sinking). Start at I and go anticlockwise.

The four quadrants of the Cartesian plane with the signs of x and y in each and example points P, Q, R and S — Slide 5: the four quadrants

How to plot a point

Start at the origin O (0, 0)
Move along the x-axis by x units (positive x: move right; negative x: move left)
From there, move parallel to the y-axis by y units (positive y: move up; negative y: move down)
Mark the point with a dot and label it (x, y)

Remember: always write x first, then y. The order matters: (3, 5) is not the same as (5, 3).

Four signposts showing the steps to plot a point, and Reiaan’s room points O, A, B, C, D1 and R1 plotted on a grid — Slide 6: how to plot a point on the Cartesian plane

Key concepts

Idea	Rule
The origin	O = (0, 0), where x = 0 and y = 0
Points on the x-axis	Of the form (x, 0): the y-coordinate is always 0. Examples: (5, 0), (−3, 0), (7.5, 0)
Points on the y-axis	Of the form (0, y): the x-coordinate is always 0. Examples: (0, 4), (0, −2), (0, 7)
(x, y) versus (y, x)	The order matters: (3, 5) ≠ (5, 3). They are equal only if x = y
Distance formula	d = √[(x₂ − x₁)² + (y₂ − y₁)²]

A summary of key coordinate geometry concepts: the origin, points on the axes, quadrant signs, order of coordinates and the distance formula — Slide 10: key concepts of coordinate geometry

Section 3 · The distance formula

Distance between two points

The key is the Baudhayana–Pythagoras theorem: in a right triangle, (hypotenuse)² = (base)² + (height)². To find the distance between two points, draw a right triangle with the horizontal shift and the vertical shift as its legs.

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

It does not matter whether (x₂ − x₁) or (y₂ − y₁) is negative, because we square both, so they become positive. A distance is always zero or more.

Worked examples

Points	Horizontal shift	Vertical shift	Distance
A (3, 4) and D (7, 1)	7 − 3 = 4	4 − 1 = 3	√(16 + 9) = √25 = 5 units
D (7, 1) and M (9, 6)	9 − 7 = 2	6 − 1 = 5	√(4 + 25) = √29 ≈ 5.39 units
M (9, 6) and A (3, 4)	9 − 3 = 6	6 − 4 = 2	√(36 + 4) = √40 ≈ 6.32 units

A triangle ADM on a grid with A (3, 4), D (7, 1) and M (9, 6), showing the horizontal and vertical shifts used in the distance formula — Slide 8: distance between two points

Three worked examples of the distance formula and practice problems, including a check whether three points lie on one line — Slide 12: distance formula, worked examples and practice

Reflection of points in the y-axis

When a shape is flipped across the y-axis, the rule is (x, y) → (−x, y): x changes sign and y stays the same.

Original	Reflected
A (3, 4)	A′ (−3, 4)
D (7, 1)	D′ (−7, 1)
M (9, 6)	M′ (−9, 6)

The distances stay the same: A′D′ = AD = 5 units, D′M′ = DM = √29 units and M′A′ = MA = √40 units. Reflection preserves distances. The shape stays exactly the same size and only its position changes (it flips left–right). This is called congruence.

Triangle ADM reflected across the y-axis to triangle A prime D prime M prime, with all the distances staying the same — Slide 9: reflection of points with negative coordinates

Section 4 · Reiaan’s room: exercises with answers

Reiaan’s room fully mapped on the coordinate plane with the bathroom, bedroom, study table, wardrobe and doors marked — Slide 11: Reiaan’s room mapped on the coordinate plane

In the room, the room door starts at D₁ (9, 0) and ends at R₁ (11.5, 0). The bathroom door lies on the y-axis from B₁ (0, 1.5) to B₂ (0, 4).

Exercise Set 1.1: the doors

Question	Answer
What are the coordinates of D₁? Is it on the x-axis?	D₁ = (9, 0). Its y-coordinate is 0, so it lies on the x-axis
How far is the door from the left wall (the y-axis)? How far from the x-axis?	9 ft from the y-axis (the left wall) and 0 ft from the x-axis, because it lies on it
If R₁ = (11.5, 0), how wide is the room door D₁R₁? Can a wheelchair enter?	Width = 11.5 − 9 = 2.5 ft = 30 inches. A standard wheelchair needs at least 32 inches, so this door is not wide enough
If B₁ (0, 1.5) and B₂ (0, 4) are the ends of the bathroom door, is it narrower or wider than the room door?	Width = 4 − 1.5 = 2.5 ft. It is exactly the same width as the room door

Exercise Set 1.2: the study table

A rectangular study table has three of its feet at (8, 9), (11, 9) and (11, 7).

The fourth foot is at (8, 7), because opposite sides of a rectangle are equal and parallel
Length = 11 − 8 = 3 ft and width = 9 − 7 = 2 ft
Whether it is a good spot is a judgement: decide it by checking that the table leaves room around the bed and the doors

Section 5 · End-of-chapter exercises with answers

Exercises Q.1 to Q.5

Question	Answer
What are the x- and y-coordinates of the point where the two axes meet?	The origin: (0, 0)
Point W has x-coordinate −5. Predict the coordinates of point H on the line through W parallel to the y-axis. In which quadrants can H lie?	H = (−5, y), where y can be any value. Since x is negative, H can lie in Quadrant II or Quadrant III (or on the x-axis if y = 0)
R (3, 0), A (0, −2), M (−5, −2) and P (−5, 2) are joined in the order R → A → M → P → R. Find perpendicular sides, a side parallel to an axis, and mirror images.	AM is horizontal (both y = −2), so it is parallel to the x-axis, and MP is vertical (both x = −5), so it is parallel to the y-axis. So AM ⊥ MP. M (−5, −2) and P (−5, 2) are mirror images of each other in the x-axis
What would a coordinate system be like if we did not have negative numbers?	Only Quadrant I could be described. Points to the left of the y-axis or below the x-axis could not be named, so we could not locate all points on a 2-D plane

Exercises Q.6 to Q.16

Question	Answer
Are M (−3, −4), A (0, 0) and G (6, 8) on one straight line? Check without plotting.	MA = √(9 + 16) = 5, AG = √(36 + 64) = 10, MG = √(81 + 144) = 15. Since MA + AG = 5 + 10 = 15 = MG, the points are collinear
M (−7, 1) is the midpoint of A (3, −4) and B (x, y). Find B.	−7 = (3 + x)/2 gives x = −17, and 1 = (−4 + y)/2 gives y = 6. So B = (−17, 6)
Show that A (1, −8), B (−4, 7) and C (−7, −4) lie on a circle centred at the origin. Find the radius.	OA = √(1 + 64) = √65, OB = √(16 + 49) = √65, OC = √(49 + 16) = √65. All are equal, so they lie on the circle, and the radius = √65
Do D (−5, 6) and E (0, 9) lie within, on or outside the same circle?	OD = √(25 + 36) = √61, which is less than √65, so D is inside. OE = √81 = 9, which is more than √65, so E is outside
Two circles on a computer screen: A has centre (100, 150) and radius 80; B has centre (250, 230) and radius 100. Do they intersect?	Distance between the centres = √(150² + 80²) = √28900 = 170 pixels. The sum of the radii is 80 + 100 = 180. Since 170 < 180, the circles intersect
Is ABCD a square for A (2, 1), B (−1, 2), C (−2, −1) and D (1, −2)?	AB = BC = CD = DA = √10, and both diagonals AC = BD = √20. All sides are equal and the diagonals are equal, so ABCD is a square with area (√10)² = 10 square units

Section 6 · Chapter summary and revision

A mind map summarising coordinate geometry: the Cartesian plane, coordinate rules, plotting a point, distance formula, reflection and important problems — Slide 13: chapter summary

Topic	Key points
Cartesian plane	Two perpendicular lines: x-axis (horizontal) and y-axis (vertical), meeting at the origin O (0, 0); the plane has 4 quadrants
Coordinates	Write (x, y), with x first. A point on the x-axis is (x, 0); a point on the y-axis is (0, y)
Quadrants	I (+, +), II (−, +), III (−, −), IV (+, −)
Plotting	Start at O; move x units along the x-axis; then y units parallel to the y-axis; mark the point
Distance	d = √[(x₂ − x₁)² + (y₂ − y₁)²] (Baudhayana–Pythagoras theorem)
Reflection in the y-axis	(x, y) → (−x, y); distances are preserved and the shape is congruent

India’s gift to coordinate geometry: the Sindhu-Sarasvati cities laid out on a grid; Baudhayana gave the distance theorem (800 BCE); Aryabhata used coordinates for the sky and sines (499 CE); Brahmagupta gave zero and negative numbers, which made the four-quadrant plane possible (628 CE).

Section 7 · Practice: multiple choice questions

Tap an option to check it at once. Open Check answer for the reason.

1The coordinates of the origin are: [1]

(a) (1, 0)(b) (1, 1)(c) (0, 0)(d) (0, 1)

Check answer

(c) (0, 0)
The origin is where the x-axis and the y-axis meet, so both coordinates are zero.

2The point (4, −3) lies in: [1]

(a) Quadrant IV(b) Quadrant II(c) Quadrant III(d) Quadrant I

Check answer

(a) Quadrant IV
x is positive and y is negative, which is Quadrant IV (+, −).

3A point on the x-axis has the form: [1]

(a) (0, 0) only(b) (x, x)(c) (0, y)(d) (x, 0)

Check answer

(d) (x, 0)
The y-coordinate of every point on the x-axis is 0.

4A point on the y-axis has the form: [1]

(a) (y, y)(b) (0, y)(c) (x, −x)(d) (x, 0)

Check answer

(b) (0, y)
The x-coordinate of every point on the y-axis is 0.

5The point (−5, 3) lies in: [1]

(a) Quadrant I(b) Quadrant III(c) Quadrant II(d) Quadrant IV

Check answer

(c) Quadrant II
x is negative and y is positive: left of the y-axis but above the x-axis.

6Which of these points lies on the y-axis? [1]

(a) (2, 2)(b) (−2, 0)(c) (−2, −2)(d) (0, −2)

Check answer

(d) (0, −2)
Its x-coordinate is 0.

7The distance between A (3, 4) and D (7, 1) is: [1]

(a) 5 units(b) 25 units(c) 7 units(d) 3 units

Check answer

(a) 5 units
√[(7 − 3)² + (1 − 4)²] = √(16 + 9) = √25 = 5.

8The distance of the point (6, 8) from the origin is: [1]

(a) 6 units(b) 10 units(c) 14 units(d) 48 units

Check answer

(b) 10 units
√(6² + 8²) = √100 = 10.

9The distance between O (0, 0) and (5, −6) is: [1]

(a) √11 units(b) √1 unit(c) 11 units(d) √61 units

Check answer

(d) √61 units
√(5² + 6²) = √(25 + 36) = √61, about 7.81 units.

10The reflection of the point (3, 4) in the y-axis is: [1]

(a) (4, 3)(b) (−3, −4)(c) (−3, 4)(d) (3, −4)

Check answer

(c) (−3, 4)
Reflection in the y-axis changes the sign of x and keeps y.

11Reflection of a triangle in the y-axis: [1]

(a) preserves all the distances, so the new triangle is congruent(b) doubles all the distances(c) changes the angles(d) halves all the distances

Check answer

(a) preserves all the distances, so the new triangle is congruent
Only the position changes; the shape stays exactly the same size.

12The distance formula is based on: [1]

(a) the theorem of Descartes(b) the Baudhayana–Pythagoras theorem(c) the rule of three(d) Euclid’s fifth postulate

Check answer

(b) the Baudhayana–Pythagoras theorem
(hypotenuse)² = (base)² + (height)² applied to the horizontal and vertical shifts.

13The points (3, 5) and (5, 3) are: [1]

(a) mirror images in the x-axis(b) both on the y-axis(c) different points(d) the same point

Check answer

(c) different points
The order of x and y matters. They are equal only when x = y.

14Which pair of signs describes Quadrant III? [1]

(a) (+, −)(b) (+, +)(c) (−, +)(d) (−, −)

Check answer

(d) (−, −)
Both coordinates are negative.

15Who formalised zero and negative numbers in 628 CE? [1]

(a) Aryabhata(b) Brahmagupta(c) Baudhayana(d) René Descartes

Check answer

(b) Brahmagupta
Without zero and negative numbers the four-quadrant plane would be impossible.

16Who made the final formalisation of 2-D coordinates in 1637? [1]

(a) René Descartes(b) Brahmagupta(c) Aryabhata(d) Baudhayana

Check answer

(a) René Descartes
That is why the system is called the Cartesian plane.

17The midpoint of (2, 4) and (6, 8) is: [1]

(a) (4, 12)(b) (8, 12)(c) (4, 6)(d) (2, 2)

Check answer

(c) (4, 6)
Midpoint = ((2 + 6)/2, (4 + 8)/2) = (4, 6).

18M (−3, −4), A (0, 0) and G (6, 8) are collinear because: [1]

(a) MA × AG = MG(b) MA = AG(c) all three have positive coordinates(d) MA + AG = MG (5 + 10 = 15)

Check answer

(d) MA + AG = MG (5 + 10 = 15)
If MA + AG = MG, the three points lie on a straight line.

19If point W has x-coordinate −5, a point H on the line through W parallel to the y-axis can lie in: [1]

(a) Quadrant II or Quadrant III(b) Quadrant I only(c) Quadrant I or Quadrant IV(d) Quadrant IV only

Check answer

(a) Quadrant II or Quadrant III
x stays −5 (negative), while y can be any value.

20The point (0, 0) lies on: [1]

(a) only the y-axis(b) both the x-axis and the y-axis(c) neither axis(d) only the x-axis

Check answer

(b) both the x-axis and the y-axis
It is the point where the two axes intersect.

ASSERTION-REASON

21Assertion (A): The points (3, 5) and (5, 3) are different points.
Reason (R): In a pair (x, y) the x-coordinate is written first, and the order matters. [1]

(a) Both A and R are true, and R is the correct explanation of A(b) Both A and R are true, but R is not the correct explanation of A(c) A is true, but R is false(d) A is false, but R is true

Check answer

(a) Both A and R are true, and R is the correct explanation of A
Both statements are true. Swapping the coordinates gives a different point unless x = y, so R explains A.

ASSERTION-REASON

22Assertion (A): The reflection of a triangle in the y-axis changes its size.
Reason (R): Reflection in the y-axis changes only the position of the triangle. [1]

(a) Both A and R are true, and R is the correct explanation of A(b) Both A and R are true, but R is not the correct explanation of A(c) A is true, but R is false(d) A is false, but R is true

Check answer

(d) A is false, but R is true
A is false: reflection preserves distances, so the triangle stays congruent. R is true.

Section 8 · Practice: short answers

EXAM QUESTION

Q1Name the four quadrants and the signs of x and y in each. [2]

Check answer

Quadrant I (+, +); Quadrant II (−, +); Quadrant III (−, −); Quadrant IV (+, −).

EXAM QUESTION

Q2Write the steps to plot a point (x, y). [3]

Check answer

Start at the origin; move x units along the x-axis (right if positive, left if negative); from there, move y units parallel to the y-axis (up if positive, down if negative); mark the point and label it (x, y).

EXAM QUESTION

Q3State the distance formula and find the distance between P (1, −8) and Q (−4, 7). [3]

Check answer

d = √[(x₂ − x₁)² + (y₂ − y₁)²]. For P and Q, the horizontal shift is −4 − 1 = −5 and the vertical shift is 7 − (−8) = 15, so d = √(25 + 225) = √250 = 5√10, about 15.81 units.

EXAM QUESTION

Q4Find the distance between O (0, 0) and Z (5, −6), and between R (7, −4) and S (0, 0). [3]

Check answer

OZ = √(25 + 36) = √61, about 7.81 units. RS = √(49 + 16) = √65, about 8.06 units.

EXAM QUESTION

Q5Reflect A (3, 4), D (7, 1) and M (9, 6) in the y-axis, and show that the distances do not change. [4]

Check answer

A′ (−3, 4), D′ (−7, 1) and M′ (−9, 6). A′D′ = √(16 + 9) = 5 = AD, D′M′ = √(4 + 25) = √29 = DM and M′A′ = √(36 + 4) = √40 = MA. So reflection preserves distances.

EXAM QUESTION

Q6Are M (−3, −4), A (0, 0) and G (6, 8) on one straight line? [3]

Check answer

MA = 5, AG = 10 and MG = 15. Since MA + AG = MG, the three points are collinear.

EXAM QUESTION

Q7M (−7, 1) is the midpoint of A (3, −4) and B (x, y). Find B. [3]

Check answer

x = 2(−7) − 3 = −17 and y = 2(1) − (−4) = 6, so B = (−17, 6).

EXAM QUESTION

Q8Show that A (1, −8), B (−4, 7) and C (−7, −4) lie on a circle centred at the origin, and find its radius. Are D (−5, 6) and E (0, 9) inside or outside? [4]

Check answer

OA = OB = OC = √65, so all three lie on the circle and the radius is √65. OD = √61 < √65, so D is inside; OE = 9 = √81 > √65, so E is outside.

EXAM QUESTION

Q9Is ABCD a square, for A (2, 1), B (−1, 2), C (−2, −1) and D (1, −2)? Find its area. [4]

Check answer

AB = BC = CD = DA = √10 and the diagonals AC = BD = √20, so ABCD is a square with area (√10)² = 10 square units.

EXAM QUESTION

Q10On a screen, circle A has centre (100, 150) and radius 80, and circle B has centre (250, 230) and radius 100. Do they intersect? [3]

Check answer

The distance between the centres is √(150² + 80²) = √28900 = 170 pixels. The sum of the radii is 180, which is more than 170, so the circles intersect.

EXERCISE

Q11In Reiaan’s room, D₁ is at (9, 0) and R₁ at (11.5, 0). How wide is the door? Can a wheelchair that needs 32 inches enter? [2]

Check answer

Width = 11.5 − 9 = 2.5 ft = 30 inches. This is less than 32 inches, so the door is not wide enough.

EXERCISE

Q12Three feet of a rectangular table are at (8, 9), (11, 9) and (11, 7). Find the fourth foot, the length and the width. [3]

Check answer

The fourth foot is at (8, 7). Length = 11 − 8 = 3 ft and width = 9 − 7 = 2 ft.

EXERCISE

Q13Why do we need negative numbers in a coordinate system? [2]

Check answer

Without negative numbers only Quadrant I could be described, so we could not locate points to the left of the y-axis or below the x-axis. With negatives, all four quadrants cover the whole plane.

EXAM QUESTION

Q14Describe India’s contribution to coordinates. [4]

Check answer

Sindhu-Sarasvati cities used north–south and east–west streets; Baudhayana (c. 800 BCE) gave the Baudhayana–Pythagoras theorem behind the distance formula; Aryabhata (c. 499 CE) used coordinates to map the sky and replaced chords with sines; Brahmagupta (c. 628 CE) formalised zero and negative numbers, which made the four-quadrant plane possible.

Section 9 · Frequently asked questions

What is the Cartesian plane?

A plane made of two perpendicular number lines, the x-axis (horizontal) and the y-axis (vertical), that meet at the origin (0, 0). It is divided into four quadrants.

Why do we write (x, y) and not (y, x)?

By convention x comes first, and the order matters: (3, 5) and (5, 3) are different points.

What is the distance formula?

d = √[(x₂ − x₁)² + (y₂ − y₁)²]. It comes from the Baudhayana–Pythagoras theorem applied to the horizontal and vertical shifts.

What are the signs in each quadrant?

Quadrant I (+, +), Quadrant II (−, +), Quadrant III (−, −) and Quadrant IV (+, −).

What is the reflection of (x, y) in the y-axis?

(−x, y). The y-coordinate stays the same and the x-coordinate changes sign.

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