MathsCBSE Class 9 · Orienting Yourself: The Use of Coordinates

The Cartesian Plane: Quadrants, Axes and Reflections, Class 9

Q: Is (0, 0) in any quadrant?

No. The origin lies on both axes and belongs to no quadrant.

Q: Why are quadrants numbered anticlockwise?

By convention, starting from the top right where both coordinates are positive.

Q: How do I find the reflection in the line y = x?

You swap the coordinates: (x, y) \to (y, x). This goes beyond the axes and origin but follows from the same idea.

Updated 17 October 2026 · Fact-checked

A point $(x, y)$ has $x$ as its signed distance from the y-axis and $y$ as its signed distance from the x-axis. Quadrant I has $(+,+)$, II has $(-,+)$, III has $(-,-)$ and IV has $(+,-)$; points on the x-axis have $y = 0$ and on the y-axis have $x = 0$. Reflecting $(x, y)$ in the x-axis gives $(x, -y)$, in the y-axis gives $(-x, y)$ and in the origin gives $(-x, -y)$.

Reflect (3, 2): the x-axis flips only y, the y-axis flips only x, and the origin flips both, moving the point to a different quadrant.

Illustration drawn for QuestionBuddy.

Understand The Cartesian Plane: Quadrants, Axes and Reflections

Axes, origin and ordered pairs. Two perpendicular number lines, the x-axis (horizontal) and the y-axis (vertical), meet at the origin $O (0, 0)$ . A point is located by an ordered pair $(x, y)$ : the first number is the x-coordinate (the signed distance from the y-axis, measured along the x-axis) and the second is the y-coordinate. Order matters: $(3, - 2)$ and $(- 2, 3)$ are different points.

Quadrants. The axes divide the plane into four quadrants, numbered anticlockwise from the top right:

Quadrant I: $x > 0$ , $y > 0$
Quadrant II: $x < 0$ , $y > 0$
Quadrant III: $x < 0$ , $y < 0$
Quadrant IV: $x > 0$ , $y < 0$

Points on the axes belong to no quadrant. A point $(a, 0)$ lies on the x-axis and $(0, b)$ on the y-axis. So $(- 7, 0)$ is on the negative x-axis and $(0, - 6)$ on the negative y-axis. Sign conditions such as $x y > 0$ (same signs: quadrant I or III) and $x y < 0$ (opposite signs: II or IV) help when only a rule is given.

Distances parallel to the axes. Two points with the same y-coordinate, $(x_{1}, k)$ and $(x_{2}, k)$ , lie on a horizontal line and are $∣ x_{1} - x_{2} ∣$ apart. Two points with the same x-coordinate are $∣ y_{1} - y_{2} ∣$ apart. The distance between $(- 6, 3)$ and $(4, 3)$ is $∣ - 6 - 4∣ = 10$ . When the distance is along a slanted line, the distance formula (next page) is needed.

Reflections. The mirror image of $P (x, y)$ in the x-axis is $(x, - y)$ , in the y-axis is $(- x, y)$ , and in the origin is $(- x, - y)$ . Reflection in an axis keeps lengths and areas unchanged, so a reflected triangle is congruent to the original. Reflecting in the x-axis and then in the y-axis gives the same point as reflection in the origin: $(x, y) \to (x, - y) \to (- x, - y)$ . The distance between $P (4, - 3)$ and its reflection in the x-axis $(4, 3)$ is $6$ , which is twice the distance of $P$ from the x-axis.

Shapes with sides parallel to the axes. If the vertices of a rectangle are $(- 2, 1)$ , $(5, 1)$ , $(5, - 3)$ and $(- 2, - 3)$ , the length is $∣5 - (- 2) ∣ = 7$ and the breadth is $∣1 - (- 3) ∣ = 4$ , so the area is $28$ . A diagonal splits it into two triangles of area $14$ each. A right triangle with its legs parallel to the axes has area $\frac{1}{2} \times horizontal leg \times vertical leg$ .

Using conditions to place a point. For $(2 k - 1, k + 3)$ : on the x-axis means $k + 3 = 0$ , so $k = - 3$ and the point is $(- 7, 0)$ ; on the y-axis means $2 k - 1 = 0$ , so $k = \frac{1}{2}$ and the point is $(0, 3.5)$ . Always substitute back to find the full point.

Maps and scales. If one unit on the grid represents $10$ m, then a gap of $10$ units is $100$ m and an area of $80$ square units is $80 \times 100 = 8000$ m $^{2}$ (the area scales with the square of the length scale).

The Cartesian Plane: Quadrants, Axes and Reflections: formulas to remember

Origin and axes: $O (0, 0); (a, 0) on x-axis; (0, b) on y-axis$; Axes belong to no quadrant.
Quadrant signs: $I (+, +), II (-, +), III (-, -), IV (+, -)$; Anticlockwise from the top right.
Horizontal distance: $∣ x_{1} - x_{2} ∣$; Same y-coordinate.
Vertical distance: $∣ y_{1} - y_{2} ∣$; Same x-coordinate.
Reflections: $(x, y) \to (x, - y), (- x, y), (- x, - y)$; In the x-axis, y-axis and origin.
Rectangle with sides parallel to axes: $A = ∣ x_{1} - x_{2} ∣ \times ∣ y_{1} - y_{2} ∣$; From two opposite corners.

How to solve The Cartesian Plane: Quadrants, Axes and Reflections questions

Read signs first, then distances, then reflections.

1For a point, state the signs of $x$ and $y$ and decide the quadrant or axis.
2For a condition such as "lies on the x-axis", set the relevant coordinate to zero and solve.
3For a distance along a line parallel to an axis, subtract the changing coordinate and take the positive value.
4For reflections, change the sign of the coordinate(s) that flip.
5For rectangles and right triangles with sides parallel to axes, find the two leg lengths from coordinates and compute the area.
6Convert grid units to real units using the scale given, remembering that areas scale with the square.

The Cartesian Plane: Quadrants, Axes and Reflections in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Reflection in the y-axis

Suppose a school map has the flagpole at the origin, one unit on the grid is 10 m, and the library stands at (4, 3). The school wants a second building symmetrically on the other side of the north-south line through the flagpole.

Where should the second building be placed, and how far apart are the two buildings?

Try it first, then show the solutionSolution

Reflection in the y-axis changes $(4, 3)$ to $(- 4, 3)$ .
The two buildings have the same y-coordinate, so the distance is $∣4 - (- 4) ∣ = 8$ units.
$8 \times 10 = 80$ m.

Answer: The second building goes at $(- 4, 3)$ , and the buildings are 80 m apart.

Memory hook: Mirror in the y-axis: flip the sign of x.

Tricks to remember The Cartesian Plane: Quadrants, Axes and Reflections

Short memory aids. Say each one aloud once, then use it on a question the same day.

x before y, anticlockwise from top right
Say "walk along, then climb" for the order $(x, y)$ , and count quadrants anticlockwise starting top right.
Use it when: Ordered pairs and quadrant names.
Zero on the axis
On the x-axis, $y = 0$ ; on the y-axis, $x = 0$ .
Use it when: Condition questions.
Mirror flips the sign
x-axis mirror flips $y$ ; y-axis mirror flips $x$ ; origin flips both.
Use it when: Reflections.

The Cartesian Plane: Quadrants, Axes and Reflections: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Locate the points · 3 marks · Short answerBoard level

State the quadrant or axis on which each point lies: $A (3, - 2)$ , $B (- 4, 5)$ , $C (0, - 6)$ , $D (- 7, 0)$ , $E (- 2, - 3)$ .

Try it first, then show the solutionSolution

$A (3, - 2)$ : $x > 0$ , $y < 0$ , quadrant IV. (½)
$B (- 4, 5)$ : $x < 0$ , $y > 0$ , quadrant II. (½)
$C (0, - 6)$ : $x = 0$ , so it lies on the y-axis (negative part). (1)
$D (- 7, 0)$ : $y = 0$ , so it lies on the x-axis (negative part). (½)
$E (- 2, - 3)$ : both negative, quadrant III. (½)

Answer: $A$ : IV; $B$ : II; $C$ : negative y-axis; $D$ : negative x-axis; $E$ : III.

Example 2: Reflections · 3 marks · Short answerBoard level

Find the reflections of $P (4, - 3)$ in the x-axis, in the y-axis and in the origin, and the distance between $P$ and its reflection in the x-axis.

Try it first, then show the solutionSolution

In the x-axis: $(4, 3)$ . (½)
In the y-axis: $(- 4, - 3)$ . (½)
In the origin: $(- 4, 3)$ . (1)
Same x-coordinate, so the distance is $∣ - 3 - 3∣ = 6$ units. (1)

Answer: $(4, 3)$ , $(- 4, - 3)$ , $(- 4, 3)$ ; the distance is 6 units.

Example 3: Rectangle on the grid · 5 marks · Long answerHarder board level

Three vertices of a rectangle with sides parallel to the axes are $(- 2, 1)$ , $(5, 1)$ and $(5, - 3)$ . Find the fourth vertex, the area, the perimeter and the length of a diagonal.

Try it first, then show the solutionSolution

The fourth vertex has the x-coordinate of $(- 2, 1)$ and the y-coordinate of $(5, - 3)$ : $(- 2, - 3)$ . (1)
Length $= ∣5 - (- 2) ∣ = 7$ , breadth $= ∣1 - (- 3) ∣ = 4$ . (1)
Area $= 7 \times 4 = 28$ square units; perimeter $= 2 (7 + 4) = 22$ units. (1½)
Diagonal $= 7^{2} + 4^{2} = 65$ units (Pythagoras). (1)
A diagonal divides the rectangle into two triangles of area $14$ each. (½)

Answer: Fourth vertex $(- 2, - 3)$ ; area $28$ ; perimeter $22$ ; diagonal $65$ .

Example 4: Points on the axes from a parameter · 3 marks · Short answerHarder board level

The point $(2 k - 1, k + 3)$ lies on the x-axis for one value of $k$ and on the y-axis for another. Find the points, and name the quadrant of the point for $k = 4$ .

Try it first, then show the solutionSolution

On the x-axis: $k + 3 = 0$ , so $k = - 3$ and the point is $(2 (- 3) - 1, 0) = (- 7, 0)$ . (1)
On the y-axis: $2 k - 1 = 0$ , so $k = \frac{1}{2}$ and the point is $(0, 3.5)$ . (1)
For $k = 4$ : $(7, 7)$ , in quadrant I. (1)

Answer: $(- 7, 0)$ for $k = - 3$ ; $(0, 3.5)$ for $k = \frac{1}{2}$ ; quadrant I for $k = 4$ .

Example 5: School grid map · 4 marks · Case-basedHOTS / competency

Case: on a school map, the flagpole is at the origin and one unit is 10 m. The library is at $(4, 3)$ , the canteen at $(- 6, 3)$ , the laboratory at $(- 6, - 5)$ and the gym at $(4, - 5)$ . (i) What shape do the four buildings form, and what are its dimensions in metres? (ii) Find the area it encloses in square metres. (iii) Is the flagpole inside the shape? (iv) The reflection of the library in the y-axis is a possible site for a notice board: where is it, and does it lie on the line joining library and canteen?

Try it first, then show the solutionSolution

(i) The sides are parallel to the axes: the horizontal sides have length $∣4 - (- 6) ∣ = 10$ units and the vertical sides $∣3 - (- 5) ∣ = 8$ units, so it is a rectangle of $100$ m by $80$ m. (1)
(ii) Area $= 10 \times 8 = 80$ square units $= 80 \times 100 = 8000$ m $^{2}$ . (1)
(iii) The x-coordinates range from $- 6$ to $4$ and the y-coordinates from $- 5$ to $3$ , so $(0, 0)$ is inside. (1)
(iv) The reflection is $(- 4, 3)$ ; it has the same y-coordinate as the library and the canteen, and $- 6 < - 4 < 4$ , so it lies on the segment joining them. (1)

Answer: (i) rectangle 100 m by 80 m; (ii) 8000 m $^{2}$ ; (iii) yes; (iv) $(- 4, 3)$ , which is on the library-canteen side.

Where students lose marks in The Cartesian Plane: Quadrants, Axes and Reflections

The slips examiners see most often, and what to write instead.

Mistake: Writing coordinates in the wrong order
Reading $(y, x)$ .
Wrong working
The point 3 units up and 2 to the right is $(3, 2)$ .
Right working
It is $(2, 3)$ : x first.
Do this: Say "along, then up".
Usually costs: The whole answer.
Mistake: Placing an axis point in a quadrant
Not seeing a zero coordinate.
Wrong working
$(0, - 6)$ lies in quadrant III.
Right working
It lies on the negative y-axis.
Do this: Check for a zero first.
Usually costs: 1 mark.
Mistake: Distance without absolute value
Subtracting in the wrong order.
Wrong working
Distance from $(- 6, 3)$ to $(4, 3)$ is $- 6 - 4 = - 10$ .
Right working
$∣ - 6 - 4∣ = 10$ .
Do this: Distances are never negative.
Usually costs: 1 mark.

Exam tips

State the signs of both coordinates in your reasoning.
Draw a quick sketch for reflection and rectangle questions.
Label axes and units on any plotted figure.
Remember that area scale factors are squared.

Practice questions on The Cartesian Plane: Quadrants, Axes and Reflections

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The distance of the point $(- 3, 4)$ from the x-axis is
(a) 3
(b) 4
(c) 5
(d) $- 4$
Hint
Distance from the x-axis is the size of the y-coordinate.
Answer
(b) 4. The distance of a point from the x-axis is $∣ y ∣ = 4$ ; from the y-axis it is $∣ x ∣ = 3$ .
Q2 · 3 marks · Short answerBoard level
A point $(a, b)$ has $ab > 0$ and $a + b < 0$ . In which quadrant does it lie? Give a reason, and name the quadrant of $(- b, a)$ .
Hint
Same signs, and a negative sum means both negative.
Answer
$ab > 0$ means $a$ and $b$ have the same sign; $a + b < 0$ means both are negative. So $(a, b)$ is in quadrant III. Then $- b > 0$ and $a < 0$ , so $(- b, a)$ has signs $(+, -)$ : quadrant IV.
Q3 · 3 marks · Short answerHarder board level
The point $P (- 3, 5)$ is reflected first in the x-axis to give $Q$ , and then $Q$ is reflected in the y-axis to give $R$ . Find $Q$ and $R$ , and show that $R$ is also the reflection of $P$ in the origin.
Hint
Reflection in the x-axis flips $y$ ; in the y-axis flips $x$ .
Answer
$Q = (- 3, - 5)$ . $R = (3, - 5)$ . The reflection of $P (- 3, 5)$ in the origin is $(3, - 5)$ , which equals $R$ .
Q4 · 5 marks · Long answerHarder board level
The vertices of a triangle are $A (2, 3)$ , $B (2, - 5)$ and $C (- 4, - 5)$ . Show that the triangle is right-angled, and find its area, the length of $A C$ and its perimeter. Write the vertices of its reflection in the y-axis.
Hint
$A B$ is vertical and $B C$ is horizontal.
Answer
$A$ and $B$ have the same x-coordinate, so $A B$ is vertical with length $∣3 - (- 5) ∣ = 8$ ; $B$ and $C$ have the same y-coordinate, so $B C$ is horizontal with length $∣2 - (- 4) ∣ = 6$ . A vertical and a horizontal line meet at a right angle, so the triangle is right-angled at $B$ . Area $= \frac{1}{2} \times 8 \times 6 = 24$ . $A C = 8^{2} + 6^{2} = 10$ . Perimeter $= 8 + 6 + 10 = 24$ . Reflection in the y-axis: $(- 2, 3)$ , $(- 2, - 5)$ , $(4, - 5)$ .
Q5 · 4 marks · Case-basedHOTS / competency
Case: a drone starts at the origin of a grid (1 unit = 1 m). It flies 8 units east, then 6 units north, then 3 units west, then 10 units south. (i) Write its position after each leg. (ii) Find the total distance flown. (iii) In which quadrant does it end? (iv) How far is its final position from where it was after the third leg?
Hint
East is $+ x$ , north is $+ y$ .
Answer
(i) $(8, 0)$ , $(8, 6)$ , $(5, 6)$ , $(5, - 4)$ . (ii) $8 + 6 + 3 + 10 = 27$ m. (iii) $(5, - 4)$ has $x > 0$ , $y < 0$ : quadrant IV. (iv) After leg 3 it was at $(5, 6)$ , directly above $(5, - 4)$ , so the distance is $∣6 - (- 4) ∣ = 10$ m.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Quadrant of $(- 3, 4)$ ?Show answer
II.
2.Reflection of $(x, y)$ in the x-axis?Show answer
$(x, - y)$ .
3.Coordinates of the origin?Show answer
$(0, 0)$ .
4.A point on the y-axis has what x-coordinate?Show answer
0.
5.Does reflection in an axis change lengths?Show answer
No.

The Cartesian Plane: Quadrants, Axes and Reflections: frequently asked questions

Is (0, 0) in any quadrant?

No. The origin lies on both axes and belongs to no quadrant.

Why are quadrants numbered anticlockwise?

By convention, starting from the top right where both coordinates are positive.

How do I find the reflection in the line y = x?

You swap the coordinates: (x, y) \to (y, x). This goes beyond the axes and origin but follows from the same idea.