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Ganita Manjari Class 9 Chapter 1, Orienting Yourself: The Use of Coordinates, explained simply

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Ganita Manjari is the new NCERT Mathematics textbook for Class 9 (Part I, first edition April 2026). Its first chapter, "Orienting Yourself: The Use of Coordinates", runs from page 1 to page 15. It teaches how two number lines at right angles let you name any point on a flat surface with two numbers, and how to measure the gap between two points with the Baudhāyana–Pythagoras theorem you met in Class 8. This guide explains each idea in plain words, works through problems from the exercises, and ends with a self-check and a way to test an AI's answer.

Keep the textbook open next to it. Our explanations and examples are our own; the textbook is the authority.

How much it counts, and which problems are examined

CBSE's Class IX Mathematics curriculum for 2026-27 makes this chapter a unit of its own: Unit III, Coordinate Geometry, worth 4 of the 80 theory marks, taught in 6 periods. The curriculum says a student should be able to:

Class 9 has no board exam: your school sets the papers, using this curriculum.

One rule from the book itself. Some problems in Ganita Manjari carry an asterisk (*). The book's "About the Book" pages (page vi) says these are for enrichment and, in its words, "not to be assessed or included in examinations"; its note to teachers says the same. In Chapter 1, end-of-chapter problems 6 to 13 are starred. They include the straight-line and midpoint problems, which the curriculum also lists, so ask your teacher how your school will treat them. They are worth doing either way: they are where the chapter's ideas get used.

How the chapter is built

1. Axes, origin and coordinates (section 1.3)

Take two number lines at right angles. The horizontal one is the x-axis, the vertical one is the y-axis, and they cross at the origin O = (0, 0). Count the same units along both. To the right of O and above O is positive; to the left and below is negative.

A point's coordinates are written (x, y), always in that order:

Points on the axes have a 0 in them: every point on the x-axis looks like (x, 0), and every point on the y-axis looks like (0, y). So the point 7 units below O is (0, −7), and the point 2.5 units to the left of O is (−2.5, 0).

Order matters. (3, 5) and (5, 3) are different points. (x, y) and (y, x) are the same point only when x = y, which is one of the "Think and Reflect" questions in the book.

2. The four quadrants

The axes cut the plane into four quadrants, numbered I to IV anticlockwise from the top right. The signs tell you which one a point is in:

QuadrantxyOur example
I (top right)++(3, 5)
II (top left)−+(−4, 7)
III (bottom left)−−(−1, −8)
IV (bottom right)+−(2, −6)

A point on an axis, such as (0, 5) or (−3, 0), is in no quadrant: it sits on the boundary.

3. Distance between two points (section 1.4)

When the two points are level or one above the other, the distance is just the difference of the coordinate that changes, taken as positive. From (2, 4) to (9, 4) is 9 − 2 = 7 units; from (−3, 4) to (5, 4) is 5 − (−3) = 8 units.

When the segment is slanted, make a right-angled triangle. Go across, then up or down. The across part and the up-or-down part are the two shorter sides, and the segment you want is the hypotenuse. By the Baudhāyana–Pythagoras theorem, the length of the segment from (x₁, y₁) to (x₂, y₂) is

√((x₂ − x₁)² + (y₂ − y₁)²)

The book works this on a triangle with corners A (3, 4), D (7, 1) and M (9, 6), and gets AD = 5 units.

Worked, our own. From P (1, 2) to Q (7, 10): across 7 − 1 = 6, up 10 − 2 = 8. Distance = √(36 + 64) = √100 = 10 units.

Worked, our own, with negatives. From (−2, 3) to (4, −5): across 4 − (−2) = 6, down 3 − (−5) = 8. Distance = √(36 + 64) = 10 units. The signs of the differences do not matter, because each one is squared.

Reflection keeps lengths. The book reflects its triangle in the y-axis: every x-coordinate changes sign, the y-coordinates stay, and all three sides keep their lengths. Try it: (4, 1) to (1, 5) is √(9 + 16) = 5, and the reflected points (−4, 1) to (−1, 5) are also 5 apart.

4. Using distance: right angles, straight lines and midpoints

Is a triangle right-angled? Find the squares of the three sides. If the two smaller squares add up to the largest, the triangle has a right angle. Our own: P (1, 1), Q (4, 5), R (8, 2).

Are three points on one straight line? (The book asks this in its starred problems 6 and 7.) One method with only what this chapter teaches: if B lies on the segment from A to C, then AB + BC = AC. Our own: A (1, 1), B (3, 4), C (7, 10). AB = √13, BC = √52 = 2√13 and AC = √117 = 3√13. Since √13 + 2√13 = 3√13, the three points are on one line.

Midpoint. The book's starred problem 9 asks you to discover the rule from examples; the curriculum lists it. The midpoint of (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2): the average of the x's and the average of the y's. Our own: the midpoint of (2, −3) and (8, 5) is (10/2, 2/2) = (5, 1).

5. Problems from the exercises, worked

Exercise Set 1.1 (pages 4-5) reads Reiaan's room from the book's Fig. 1.3, so you need the figure in front of you. Here are four problems that can be worked from the text alone, plus one starred problem. The working is ours.

Exercise Set 1.2, problem 1 (page 8). Three feet of a rectangular table are at (8, 9), (11, 9) and (11, 7). Where is the fourth? The first two share y = 9, the last two share x = 11, so the fourth must share x = 8 with the first and y = 7 with the last: (8, 7). The table is 11 − 8 = 3 units one way and 9 − 7 = 2 units the other. Its height cannot be read from a floor plan: the plan is flat and shows only length and width.

End-of-chapter problem 3 (page 12). R (3, 0), A (0, −2), M (−5, −2) and P (−5, 2) are joined in order.

End-of-chapter problem 15 (page 14). A screen is 800 pixels wide and 600 high, with the origin at the bottom left. Circle A has centre (100, 150) and radius 80; circle B has centre (250, 230) and radius 100.

End-of-chapter problem 16 (page 14). A (2, 1), B (−1, 2), C (−2, −1) and D (1, −2). Each side squared is 3² + 1² = 10, so all four sides are √10. Each diagonal squared is 20, and 10 + 10 = 20, so every corner is a right angle. ABCD is a square, with area (√10)² = 10 square units.

A starred one, problem 12 (page 13). Do A (1, −8), B (−4, 7) and C (−7, −4) lie on a circle K with centre O (0, 0)? Their squared distances from O are 1 + 64, 16 + 49 and 49 + 16, all 65. So yes, and K has radius √65. For D (−5, 6) the squared distance is 61, less than 65, so D is inside K. For E (0, 9) it is 81, more than 65, so E is outside.

6. Five things that cost marks

  1. Swapping the order. (x, y) is across first, then up. (2, −6) and (−6, 2) are in different quadrants.
  2. Putting an axis point in a quadrant. (0, 5) is on the y-axis, not in Quadrant I.
  3. Dropping a minus inside the bracket. From −2 to 4 is 4 − (−2) = 6, not 2.
  4. Forgetting the square root. √(36 + 64) is 10. Stopping at 100 gives the square of the distance.
  5. Adding the distances across and up. From (1, 2) to (7, 10) is not 6 + 8 = 14 units; the straight line is 10.

7. Self-check: 6 questions with answers

Try them on paper first. Answers are below.

  1. In which quadrant is (−6, 2)?
  2. Write the coordinates of the point on the y-axis 7 units below the origin.
  3. How far apart are (−3, 4) and (5, 4)?
  4. Find the distance between (1, −2) and (4, 2).
  5. A triangle has corners (0, 0), (6, 0) and (0, 8). Is it right-angled? How long is its longest side?
  6. (Starred topic in the book.) Find the midpoint of (−4, 6) and (10, −2).

Answers: 1) Quadrant II (x negative, y positive). 2) (0, −7). 3) 8 units: same y, and 5 − (−3) = 8. 4) Across 3, up 4: √(9 + 16) = 5 units. 5) Yes, the right angle is at (0, 0), where the two sides lie along the axes. The longest side runs from (6, 0) to (0, 8): √(36 + 64) = 10 units. 6) ((−4 + 10)/2, (6 + (−2))/2) = (3, 2).

8. How to check an AI's answer for this chapter

More on using AI well: how to use ChatGPT for homework without cheating yourself.

Practise this chapter

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Sources

All checked on 21 October 2026.

The section names, page numbers, problem numbers, the starred-problem rule and the curriculum facts are from the documents above. All explanations, worked examples and self-check questions on this page are our own, and every number was checked by calculation (49 checks). Our working for the book's problems is ours, not NCERT's.

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