Class 9 › Mathematics › Chapter

Coordinates (The Cartesian Plane)

Axes, origin, ordered pairs, quadrants, distance formula, reflection, midpoint and trisection points, and how coordinates are used to test geometric properties.

1. Coordinate System

Coordinate system: a structured framework (like the grid lines of a map or graph paper) that uses numbers to describe the exact position of a point or object. The numbers are the coordinates of the point.

1-D (number line)

One number locates a point. Used for integers, rationals and decimals.

2-D (plane)

Two numbers locate a point, using two perpendicular number lines. This is the Cartesian coordinate system.

Key idea (Descartes): every point in a plane is fixed by exactly two numbers, its distances from two perpendicular axes.
Why it matters: shapes and curves can be described by algebra and equations, and equations can be seen as shapes, which joins geometry and algebra.
Role of negative numbers: the origin is zero and the negative axes represent values less than zero. Without negatives only one quarter of the plane (Quadrant I) could be located.

2. Axes, Origin and Sign Convention

x-axis

The horizontal number line.

y-axis

The vertical number line.

Origin $O$

The point where the axes intersect (cross at $90^\circ$). Its coordinates are $(0, 0)$.

Sign Convention

Equal units (same scale) are marked on both axes, measured from $O$.
Positive: to the right of $O$ along the x-axis, and upwards from $O$ along the y-axis.
Negative: to the left of $O$ along the x-axis, and downwards from $O$ along the y-axis.

"Axes" is the plural of "axis". Position of the origin is a convention; for example, computer screens place the origin at the bottom-left corner.

3. Ordered Pair and Coordinates of a Point

Coordinates of a point $P$ $$P = (x, y) \quad\text{or simply}\quad P(x, y)$$

Meaning of $x$ and $y$:
• x-coordinate (abscissa) = perpendicular distance of $P$ from the y-axis, measured along the x-axis.
• y-coordinate (ordinate) = perpendicular distance of $P$ from the x-axis, measured along the y-axis.

$(x, y)$ is an ordered pair: the x-coordinate is always written first.
Two ordered pairs are equal only if both corresponding coordinates are equal.
$(x, y) = (y, x)$ if and only if $x = y$. If $x \neq y$, then $(x, y) \neq (y, x)$ (they are different points).
Each point has exactly one ordered pair, and each ordered pair marks exactly one point.

4. Points on the Axes and Lines Parallel to Them

On the x-axis

$(x, 0)$

y-coordinate is $0$. If $x \gt 0$ the point is right of $O$; if $x \lt 0$, left of $O$.

On the y-axis

$(0, y)$

x-coordinate is $0$. If $y \gt 0$ the point is above $O$; if $y \lt 0$, below $O$.

Origin

$(0, 0)$

Both coordinates are $0$. It is the only point lying on both axes.

Parallel to y-axis

$x = k$

Every point on a vertical line has the same x-coordinate $k$: $(k, y)$.

Parallel to x-axis

$y = k$

Every point on a horizontal line has the same y-coordinate $k$: $(x, k)$.

5. Cartesian Plane and Quadrants

The plane in which the axes lie is called the Cartesian plane, coordinate plane or xy-plane. The two axes divide it into four parts called quadrants, numbered I to IV anticlockwise, starting from the top right.

The four quadrants and the signs of $(x, y)$ in each.

Quadrant	x-coordinate	y-coordinate	Location
I	positive	positive	Top right
II	negative	positive	Top left
III	negative	negative	Bottom left
IV	positive	negative	Bottom right

Note: points lying on an axis (including the origin) do not belong to any quadrant.

6. Distance Between Points Along an Axis Direction

Horizontal segment (same $y$)

$$d = |x_2 - x_1|$$

For points $(x_1, y)$ and $(x_2, y)$.

Vertical segment (same $x$)

$$d = |y_2 - y_1|$$

For points $(x, y_1)$ and $(x, y_2)$.

The absolute value $|\cdot|$ keeps the distance positive, so the order of subtraction does not matter.
This applies whether the points lie on an axis or on any line parallel to an axis.

7. Distance Formula

When the segment joining two points is not parallel to either axis, we form a right-angled triangle and apply the Baudhāyana–Pythagoras Theorem (hypotenuse$^2$ = sum of squares of the other two sides).

Triangle $ACB$ is right-angled at $C$, so $AB$ is the hypotenuse.

Derivation

1. Let $A(x_1, y_1)$ and $B(x_2, y_2)$. Draw a vertical line through $A$ and a horizontal line through $B$, meeting at $C(x_1, y_2)$. Then $\angle C = 90^\circ$.

2. Horizontal side $BC = |x_2 - x_1|$ and vertical side $AC = |y_2 - y_1|$ (Section 6).

3. By the Baudhāyana–Pythagoras Theorem: $AB^2 = BC^2 + AC^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$.

4. Taking the positive square root gives the distance formula.

Distance Formula $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ Distance of $P(x, y)$ from the Origin $$OP = \sqrt{x^2 + y^2}$$

Procedure: subtract the x-coordinates, subtract the y-coordinates, square both, add, then take the square root.
Sign does not matter: squaring removes any negative sign, so the formula works for points in any quadrant and the order of the two points can be swapped.
Distance is never negative, and $d = 0$ only when the two points coincide.
If the segment is parallel to an axis, the formula reduces to $|x_2 - x_1|$ or $|y_2 - y_1|$.

8. Reflection in the Axes

The mirror image of a point is found by treating an axis as a mirror. A point and its image are at the same perpendicular distance from the mirror axis, on opposite sides.

Reflection of $P(x, y)$ in the y-axis gives $P'(-x, y)$.

In the y-axis

$$(x, y) \to (-x, \; y)$$

The x-coordinate changes sign.

In the x-axis

$$(x, y) \to (x, \; -y)$$

The y-coordinate changes sign.

Properties of reflection:
• Lengths are preserved: the sides of a figure and its image are equal, so shape and size stay the same (verified with the distance formula).
• What changes: the position of the figure and the sign of one coordinate.
• Points on the mirror axis are their own images.
• Two points that are mirror images in an axis share one coordinate and have opposite values of the other.

9. Midpoint Formula

If $M$ is the midpoint of segment $ST$ with $S(x_1, y_1)$ and $T(x_2, y_2)$, each coordinate of $M$ is the average of the corresponding coordinates of $S$ and $T$.

Midpoint $$M = \left(\frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2}\right)$$

Test for a midpoint: $M$ is the midpoint of $ST$ only if both averages match $M$'s coordinates.
Finding an endpoint: if $M(m_1, m_2)$ and $S(x_1, y_1)$ are known, the other end is $T = (2m_1 - x_1, \; 2m_2 - y_1)$.
Triangle from midpoints: if $D, E, F$ are the midpoints of $BC, CA, AB$, then $A = E + F - D$, $B = D + F - E$, $C = D + E - F$ (coordinate-wise).

10. Points of Trisection of a Segment

The points $P$ and $Q$ that divide $AB$ into three equal parts ($P$ nearer to $A$, $Q$ nearer to $B$) for $A(x_1, y_1)$ and $B(x_2, y_2)$:

Trisection Points $$P = \left(\frac{2x_1 + x_2}{3}, \; \frac{2y_1 + y_2}{3}\right), \qquad Q = \left(\frac{x_1 + 2x_2}{3}, \; \frac{y_1 + 2y_2}{3}\right)$$

Idea: $P$ is $\tfrac13$ of the way from $A$ to $B$, i.e. $P = A + \tfrac13(B - A)$, and $Q$ is $\tfrac23$ of the way, i.e. $Q = A + \tfrac23(B - A)$.
Equivalent method: $Q$ is the midpoint of $PB$ and $P$ is the midpoint of $AQ$.
Ratio form: $AP : PQ : QB = 1 : 1 : 1$.

11. Applications of Coordinates

Collinear points

Three points on one line

Find the three distances. The points are collinear when the largest distance equals the sum of the other two. No plotting is needed.

Triangles

Type of triangle

Compare side lengths: two equal means isosceles, all three equal means equilateral. If $a^2 + b^2 = c^2$ for the sides, it is right-angled.

Quadrilaterals

Square / Rectangle

Square: four equal sides and equal diagonals; area = side$^2$. Rectangle: opposite sides equal and diagonals equal.

Circle (centre O)

Position of a point

For radius $r$: $x^2 + y^2 = r^2$ means on the circle, $\lt r^2$ means inside, $\gt r^2$ means outside.

Two circles

Do they intersect?

With centre distance $d$ and radii $r_1, r_2$: they intersect if $|r_1 - r_2| \lt d \lt r_1 + r_2$; touch if $d = r_1 + r_2$ or $d = |r_1 - r_2|$.

Within a boundary

Fits inside a region?

A circle of radius $r$ and centre $(a, b)$ lies inside a rectangle if $a \pm r$ and $b \pm r$ stay within the rectangle's limits.

Predicting from coordinates

Parallel & perpendicular sides

Same y-coordinates: side parallel to the x-axis. Same x-coordinates: parallel to the y-axis. These two types of sides are perpendicular.

12. Quick Revision Sheet

Concept	Result
Origin	$(0, 0)$
Point on x-axis / y-axis	$(x, 0)$ / $(0, y)$
Quadrant signs	I $(+,+)$, II $(-,+)$, III $(-,-)$, IV $(+,-)$
$(x, y) = (y, x)$	only when $x = y$
Distance along an axis direction	$\|x_2 - x_1\|$ or $\|y_2 - y_1\|$
Distance formula	$\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Distance from origin	$\sqrt{x^2 + y^2}$
Reflection in y-axis / x-axis	$(-x, y)$ / $(x, -y)$
Midpoint	$\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
Trisection points	$\left(\frac{2x_1 + x_2}{3}, \frac{2y_1 + y_2}{3}\right)$ and $\left(\frac{x_1 + 2x_2}{3}, \frac{y_1 + 2y_2}{3}\right)$

Common mistakes:
• Writing $(y, x)$ instead of $(x, y)$.
• Treating a point on an axis as lying in a quadrant.
• Mishandling signs: $(-a)^2 = a^2$, not $-a^2$.
• Forgetting the final square root in the distance formula.