Class 9 Maths Chapter 1: The Use of Coordinates, Notes

Class 9 Maths chapter 1 in the new textbook is Orienting Yourself: The Use of Coordinates. In this guide you will learn how the Cartesian plane works, how to read and plot points, which quadrant a point lies in, how to find the distance between two points, and what happens to a point when it is reflected in an axis. Solved important questions come at the end.

Where the idea of a grid came from

The chapter begins with history. The Sindhu-Sarasvati Civilisation planned its cities on grids, with streets running North-South and East-West about 10 metres apart, so a building could be found by counting units from a central point.

Indian mathematicians built the ideas behind coordinates:

• Baudhāyana (c. 800 BCE) used perpendicular lines in geometric constructions and gave the Baudhāyana-Pythagoras Theorem.
• Ujjayinī served as the zero-longitude reference meridian.
• Āryabhaṭa (c. 499 CE) introduced sines in place of Greek chords, making coordinate calculations for the sky and the Earth easier.
• Brahmagupta (c. 628 CE) formalised zero and negative numbers as algebraic entities.

These ideas travelled to the Arab world. Al-Bīrūnī (c. 1000 CE) used Indian trigonometric methods to work out the coordinates of cities, and Ömar Khayyām (c. 1100 CE) solved algebra problems using geometry. René Descartes formalised the Cartesian system in 1637 CE.

The Cartesian plane

Two perpendicular number lines locate any point on a flat surface.

• The x-coordinate is the perpendicular distance of the point from the y-axis.
• The y-coordinate is the perpendicular distance of the point from the x-axis.
• Right of the origin and upwards are positive; left and downwards are negative.

The plane is called the Cartesian plane, the coordinate plane or the xy-plane.

Quadrants and signs

The axes divide the plane into four quadrants, numbered I, II, III and IV anticlockwise, starting from the top right.

A point on an axis belongs to no quadrant. For example, (5, 0) is on the x-axis, not in Quadrant I or IV.

Points on the axes

• On the x-axis, y = 0. Such points look like (x, 0).
• On the y-axis, x = 0. Such points look like (0, y).

Example: A(4.5, 0) is on the x-axis, 4.5 units right of the origin. B(0, -4.5) is on the y-axis, 4.5 units below the origin.

Order matters. (x, y) and (y, x) are the same point only when x = y. Otherwise they are different points. So (2, 7) and (7, 2) are not the same.

Distance along lines parallel to the axes

• Same y-coordinate (line parallel to the x-axis): distance = |x2 - x1|.
• Same x-coordinate (line parallel to the y-axis): distance = |y2 - y1|.

We use the absolute value because a distance is a length and cannot be negative.

Example: P(2, 5) and Q(2, -3) share x = 2. Distance = |5 - (-3)| = |5 + 3| = 8 units.

The distance formula

For any two points A(x1, y1) and D(x2, y2), draw a right-angled triangle. Its horizontal side is the change in x, its vertical side is the change in y, and AD is the hypotenuse. By the Baudhāyana-Pythagoras Theorem:

AD² = (x2 - x1)² + (y2 - y1)², so AD = √((x2 - x1)² + (y2 - y1)²)

Whether a difference is positive or negative does not matter, because squaring makes it non-negative.

Example: A(3, 4) and D(7, 1).

• Change in x = 7 - 3 = 4
• Change in y = 1 - 4 = -3
• AD = √(4² + (-3)²) = √(16 + 9) = √25 = 5 units

Example: from (0, 0) to (6, 8): √(36 + 64) = √100 = 10 units.

Reflections in the axes

A reflection makes a mirror image of a point across an axis.

• Reflect (x, y) in the y-axis: (-x, y). The x-coordinate changes sign.
• Reflect (x, y) in the x-axis: (x, -y). The y-coordinate changes sign.

Reflection is an isometry: it keeps distances the same. A reflected triangle has sides of exactly the same lengths as the original.

Example: (5, -2) reflected in the x-axis becomes (5, 2).

Coordinates in real life

• City planning: the intersection (2, 5) could mean where the 2nd North-South street meets the 5th East-West street.
• Computer graphics: in the chapter's example, a screen is 800 pixels wide and 600 pixels high, with (0, 0) at the bottom-left. Shapes are drawn from their centre coordinates and size.
• Interior design: if you know three corners of a rectangular table, you can find the fourth, because opposite sides are equal and parallel.

Example: on the 800 by 600 screen, does a circle with centre (100, 150) and radius 80 stay on screen? Leftmost point: 100 - 80 = 20. Lowest point: 150 - 80 = 70. Both are above 0, and the right and top edges (180 and 230) are well inside 800 and 600. So yes.

Remember this

• Quadrants go anticlockwise: I (+, +), II (-, +), III (-, -), IV (+, -).
• x-axis points: (x, 0). y-axis points: (0, y). Points on axes are in no quadrant.
• Distance formula: √((x2 - x1)² + (y2 - y1)²).
• Reflection in x-axis flips the sign of y; in y-axis, the sign of x.

Important questions with answers

1. In which quadrant does (-4, -7) lie? Both coordinates are negative, so Quadrant III.

2. Where does (0, -6) lie? x = 0, so it is on the y-axis, 6 units below the origin. It is in no quadrant.

3. What does the x-coordinate of a point tell you? Its perpendicular distance from the y-axis.

4. Find the distance between (1, 2) and (7, 2). Same y-coordinate, so |7 - 1| = 6 units.

5. Find the distance between (1, 1) and (4, 5). √((4 - 1)² + (5 - 1)²) = √(9 + 16) = √25 = 5 units.

6. Reflect (-3, 4) in the y-axis. Change the sign of x: (3, 4).

7. Who formalised zero and negative numbers as algebraic entities? Brahmagupta (c. 628 CE).

8. Are (3, 8) and (8, 3) the same point? No. They are the same only when the two coordinates are equal.

Common mistakes to avoid

• Swapping x and y when plotting. x always comes first.
• Putting a point like (5, 0) in a quadrant. Points on axes are in no quadrant.
• Writing a negative distance. Use the absolute value.
• Changing both signs in a reflection. Only one coordinate changes.

For more practice with plotting points and distances, study this chapter with Joy on Learnijoy.