Class 9 Maths Chapter 1: Use of Coordinates, Notes and Formulas

These class 9 maths chapter 1 notes explain “Orienting Yourself: The Use of Coordinates”, the first chapter of the new NCERT book Ganita Manjari, Part 1. The chapter shows how two numbers can fix any point on a flat surface. Therefore, first, learn the plane and the four quadrants. Then practise plotting, reflection and the distance and midpoint formulas, since they solve most questions.

Class 9 Maths Chapter 1: The Cartesian Plane

The Cartesian plane has two number lines that cross at right angles. The horizontal line is the x-axis, and the vertical line is the y-axis. They meet at the origin, which is written O and has coordinates (0, 0). Both axes use the same scale, so numbers are marked outward from the origin.

A point is described by an ordered pair (x, y). Here, x is the abscissa, which tells you how far to move along the x-axis. Next, y is the ordinate, which tells you how far to move up or down. For P(a, b), a is the abscissa and b is the ordinate.

Quadrants and Signs

The axes divide the plane into four quadrants. Because of this, their signs follow a fixed pattern.

Quadrant	Sign of x	Sign of y	Example
I	Positive	Positive	(4, 5)
II	Negative	Positive	(−2, 3)
III	Negative	Negative	(−3, −1)
IV	Positive	Negative	(6, −2)

However, points on the axes belong to no quadrant. A point on the x-axis looks like (x, 0), and a point on the y-axis looks like (0, y).

How to Plot a Point

To plot P(a, b), start at the origin. First, move |a| units along the x-axis, to the right if a is positive and to the left if it is negative. Then move |b| units up if b is positive, or down if it is negative. Finally, mark the point and label it.

For instance, to plot (−3, 2), move 3 units left and then 2 units up. This lands in Quadrant II.

Reflection in the Axes

A reflection is a mirror image. When you reflect a point in the x-axis, the x-coordinate stays the same and the y-coordinate changes sign. So (3, 4) becomes (3, −4). When you reflect in the y-axis, the y-coordinate stays the same and the x-coordinate changes sign. So (3, 4) becomes (−3, 4).

Distance Between Two Points

Take points P(x₁, y₁) and Q(x₂, y₂). Next, draw a right triangle with PQ as its hypotenuse. The horizontal side is |x₂ − x₁|, and the vertical side is |y₂ − y₁|. By the Pythagoras theorem, PQ = √[(x₂ − x₁)² + (y₂ − y₁)²].

Also, the distance of a point (x, y) from the origin is a special case: √(x² + y²).

Example 1. Find the distance between A(1, 2) and B(4, 6).

AB = √[(4 − 1)² + (6 − 2)²] = √(9 + 16) = √25 = 5 units.

Midpoint of a Line Segment

The midpoint divides a segment in the ratio 1 : 1. Therefore, for P(x₁, y₁) and Q(x₂, y₂) it is ((x₁ + x₂)/2, (y₁ + y₂)/2). In other words, you average the x-values and then average the y-values.

Example 2. Find the midpoint of (2, −3) and (8, 5).

First, the x-value is (2 + 8)/2 = 5. Then the y-value is (−3 + 5)/2 = 1. Hence, the midpoint is (5, 1).

Collinear Points and Shapes

Points are collinear when they lie on one straight line. To check three points A, B and C, find the three distances. If the longest distance equals the sum of the other two, the points are collinear.

Coordinates also help you test shapes, so they link algebra with geometry. A rectangle has equal opposite sides and equal diagonals. Likewise, a rhombus has four equal sides. Finally, in a parallelogram, the diagonals share the same midpoint. Therefore, the distance and midpoint formulas let you prove a shape’s type without drawing it.

Example 3. Show that A(0, 0), B(2, 2) and C(4, 4) are collinear.

Here AB = √8 and BC = √8, while AC = √32 = 2√8. Since AB + BC = AC, the points are collinear.

Common Mistakes in Coordinate Geometry

• Students swap x and y, so they often plot the point in the wrong place.
• Similarly, many forget the sign when they subtract negative numbers inside the distance formula.
• Also, some forget the square root at the end, so they report the squared distance.
• Finally, others call a point on an axis a quadrant point, although it belongs to no quadrant.

How to Study This Chapter

First, draw a graph sheet for every problem, even if it is quick. Next, learn the quadrant table and the two formulas. After that, solve one example of each type: plotting, reflection, distance, midpoint and collinearity. Finally, check your answers by sketching the points. For the previous subject chapters, read class 9 science chapter 1 and motion class 9 notes. You can also browse the Class 9 Maths category and the Class 9 notes hub. Older students can see how coordinates connect to trigonometry class 10 notes.

Wrap-Up

Class 9 maths chapter 1 gives you a tool you will use in every later geometry chapter. So learn the quadrant signs and the two formulas, and sketch each problem. Then continue with the Class 9 Maths category, or return to the The Note homepage.