Coordinate Geometry Class 9: Cartesian System Explained

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What Is Coordinate Geometry? 

Coordinate geometry is a branch of geometry that deals with the position of a point. Every point has a position, and that position is described using coordinates. On a graph, this is shown using an x-axis and a y-axis that are mutually perpendicular to each other, with numbers marked as positive on one side and negative on the other, both for the x-axis and the y-axis.

This idea is the same as being told to sit in the "third row, fourth seat" in an exam hall, or marking a position in a video game — a location is fixed by two numbers together, not one.

The Historical Background of Coordinate Geometry 

The chapter traces coordinate geometry back through several civilisations and mathematicians:

• Indus-Saraswati Civilisation — credited with the first use of grid city streets, laid out roughly 10 metres apart.
• Baudhayana — used north-south and east-west lines for geometric constructions.
• Ujjain — used as the central longitude (meridian) of the globe for navigation.
• Brahmagupta — formalised the use of zero and negative numbers.
• René Descartes — the mathematician credited with developing the full two-dimensional coordinate system, which is why it is called the Cartesian system.

💡 How the idea came about: According to the chapter, Descartes was lying down and noticed a fly moving to different positions on the ceiling. Watching the fly's changing position is described as the spark that led him to work out coordinate geometry.

Two dimensions means a point can move in two directions (for example, left-right and up-down). If a point can also move in a third direction, that becomes three-dimensional geometry.

Understanding the Cartesian System 

In the Cartesian system, a plane has an x-axis and a y-axis, which are perpendicular to each other. The point where they meet is called the origin.

• The x-coordinate is called the abscissa — the distance measured along the x-axis.
• The y-coordinate is called the ordinate — the distance measured along the y-axis.
• A point is written as an ordered pair in the form (x, y) — always x first, then y. Writing it as (y, x) is incorrect. If a third axis (z) is involved, the point is written as (x, y, z).

The x-axis and y-axis together divide the plane into four parts, called quadrants.

The Four Quadrants and Their Signs 

Using these rules, points can be placed directly without drawing a graph:

• (−5, 3) → x is negative, y is positive → Second Quadrant
• (3, −5) → x is positive, y is negative → Fourth Quadrant
• (0, −4.5) → x is 0, so the point lies on the y-axis
• (4.5, 0) → y is 0, so the point lies on the x-axis

Two related rules follow directly from this:

1. Any point on the x-axis has its y-coordinate equal to 0.
2. Any point on the y-axis has its x-coordinate equal to 0.

There is also an equality rule for ordered pairs: (x, y) and (y, x) represent the same point only when x = y. For example, (3, 3) written either way is the same point, but (3, 2) and (2, 3) are two different points.

Solving Position-Based Questions Using a Room Diagram 

CBSE has started including picture-based questions on coordinate geometry, and the chapter works through one using a room layout — a rectangle with corners at O(0,0), (0,10), (12,0), and (12,10).

• D1 marks the door of the room. It is 8 units from the left wall (the y-axis) and 0 units from the x-axis, since the door lies exactly on the x-axis. So the coordinates of D1 are (8, 0).
• R1, given as the point (11.5, 0), marks the other end of the door.
• Door width = distance between D1 and R1 = 11.5 − 8 = 3.5 units.
• The bathroom door, marked by B1(0, 1.5) and B2(0, 4), has a width of 4 − 1.5 = 2.5 units — narrower than the main room door (3.5 units), which matches what you would expect in a real room.

A similar picture-based question uses a study table with corners T1(8,9), T2(11,9), and T3(11,7):

• Since T1 and T2 share the same y-coordinate (9), the line joining them is horizontal, and the fourth corner (T4) must also have y = 7 to match T3.
• This gives T4 = (8, 7).
• Length of the table = 11 − 8 = 3 feet.
• Width of the table = 9 − 7 = 2 feet.

The general rule from these examples: on a horizontal line (parallel to the x-axis), all points share the same y-coordinate. On a vertical line (parallel to the y-axis), all points share the same x-coordinate.

The Distance Formula

The distance formula is the most important formula in this chapter, and CBSE sets at least one question from it every year. For two points (x₁, y₁) and (x₂, y₂), the distance between them is:

Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]

This comes from the Pythagoras theorem, using the horizontal and vertical gaps between the two points as the two legs of a right triangle.

Worked examples from the chapter:

• Distance of point (4, 0) from the x-axis = 0, because the point already lies on the x-axis.
• For a point like (−1, 7), its distance from the x-axis is 7 units. A common mistake is answering with a negative number — remember that distance can never be negative, so any negative result should be written as its positive value.
• Distance of (−6, 8) from the origin (0,0): using the formula, (−6−0)² + (8−0)² = 36 + 64 = 100, and √100 = 10. So the distance is 10 units (not −10, since distance is never negative).
• Distance between P(−2, 3) and Q(4, −1): (4−(−2))² + (−1−3)² = 36 + 16 = 52. So PQ = √52 units.
• Distance between A(−6, 7) and B(−1, −5): (−1−(−6))² + (−5−7)² = 25 + 144 = 169, and √169 = 13. So AB = 13 units.

Checking If Points Form a Square 

This is a well-known NCERT back-exercise question: plot A(2,1), B(−1,2), C(−2,−1), and D(1,−2), and check whether ABCD is a square.

Plotting alone can make it look like a square, but the chapter is clear that you must verify it using the distance formula — a shape can look square but actually be a rhombus, since a rhombus also has all four sides equal.

Step 1 — Check all four sides:

All four sides are equal — but this alone only proves it could be a rhombus.

Step 2 — Check the diagonals (a square must have equal diagonals, unlike a rhombus):

• AC = √[(−2−2)² + (−1−1)²] = √(16+4) = √20 units
• BD = √[(1−(−1))² + (−2−2)²] = √(4+16) = √20 units

Since all sides are equal and both diagonals are equal, ABCD is confirmed to be a square.

Area of the square = side² = (√10)² = 10 square units.

This full method — all sides equal, then both diagonals equal — is worth remembering, since this exact type of question is typically worth a minimum of three marks in periodic tests.

Reflection of a Point in the Axes

Reflection works the same way a mirror does: an image forms at the same distance on the opposite side of the mirror line.

Reflection along the y-axis (mirror placed vertically): the y-coordinate stays the same, but the sign of the x-coordinate flips.

• Point (3, 4) in the first quadrant reflects to (−3, 4) in the second quadrant.
• Point (3, −2) in the fourth quadrant reflects to (−3, −2) in the third quadrant.

Reflection along the x-axis (mirror placed horizontally, like a floor): the x-coordinate stays the same, but the sign of the y-coordinate flips.

• Point (2, 4) reflects to (2, −4).

Collinear Points 

Points that lie on the same straight line are called collinear points — like three buttons lined up in a single row on a phone.

To prove three points A, B, and C are collinear using the distance formula, calculate AB, BC, and AC. If AB + AC = BC (or one distance equals the sum of the other two, depending on which point lies in the middle), the three points are collinear.

In the chapter's worked example, the three distances came out to AB = 3√5 units, AC = 2√5 units, and BC = 5√5 units. Since 3√5 + 2√5 = 5√5, the condition AB + AC = BC is satisfied — proving the three points are collinear.

The Midpoint Formula

The midpoint is the point exactly halfway between two other points — like the centre point of a rope tied at both ends to two equal lengths.

For two points (x₁, y₁) and (x₂, y₂), the midpoint formula is:

Midpoint = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

Worked examples:

• Midpoint of (−3, 9) and (−6, −4): ((−3−6)/2, (9−4)/2) = (−9/2, 5/2) — this exact question was asked by CBSE in the 2021 board exam.
• Midpoint of P(−4, 5) and Q(4, 6): ((−4+4)/2, (5+6)/2) = (0, 5.5). Since the x-coordinate is 0, this midpoint lies on the y-axis.
• Dividing a segment into four equal parts: for A(−2, 2) and B(2, 8), the midpoint Q of AB is (0, 5). P, the midpoint of A and Q, is (−1, 3.5). R, the midpoint of Q and B, is (1, 6.5). Together, P, Q, and R divide AB into four equal parts.

A related CBSE-style question uses the distance formula with a circle: for a circle of radius 3 units centred at the origin, a point lies outside the circle if its distance from the origin is greater than 3 units — checked the same way, point by point, using the distance formula.

A picture-based campaign-board question follows the same pattern: for a rectangle with corners A(1,1) and C(7,5), the diagonal AC = √[(7−1)² + (5−1)²] = √(36+16) = √52 units, the sides AB and CD measure 6 units, the sides BC and AD measure 4 units, and the area of the rectangle works out to 6 × 4 = 24 square units.