MathsICSE Class 10 · Reflection

Reflection in the x-axis, y-axis and the Origin: Rules and Graph Questions (ICSE Class 10)

Updated 15 October 2026 · Fact-checked

Reflection in the x-axis changes the sign of the y-coordinate: (x, y) → (x, −y). Reflection in the y-axis changes the sign of x: (x, y) → (−x, y). Reflection in the origin changes both: (x, y) → (−x, −y), and it equals reflecting in one axis and then the other.

Notice that the x-axis flips only y, the y-axis flips only x, and the origin flips both, so P's image lands in another quadrant.

Illustration drawn for QuestionBuddy.

Understand Reflection in the Axes and the Origin

A reflection in a line (the mirror line) sends each point $P$ to its image $P^{'}$ so that the mirror line is the perpendicular bisector of $P P^{'}$ : $P^{'}$ is as far behind the mirror as $P$ is in front, on the same perpendicular.

The three rules.

x-axis ( $M_{x}$ ): $(x, y) \to (x, - y)$ . The point flips up or down; $x$ stays.
y-axis ( $M_{y}$ ): $(x, y) \to (- x, y)$ . The point flips left or right; $y$ stays.
Origin ( $M_{O}$ ): $(x, y) \to (- x, - y)$ . The origin is the midpoint of $P P^{'}$ , so the point goes 'through the origin' to the opposite quadrant.

A quick test: the coordinate that measures distance from the mirror line changes sign. The x-axis is the line $y = 0$ , so $y$ changes; the y-axis is $x = 0$ , so $x$ changes.

Combining reflections. Reflecting in the x-axis and then the y-axis (in either order) gives $(x, y) \to (x, - y) \to (- x, - y)$ , the same as $M_{O}$ . Similarly $M_{x}$ followed by $M_{O}$ equals $M_{y}$ . Reflecting twice in the same line brings the point back.

Working backwards. If the image is given, apply the same reflection again: $M_{x}$ undoes itself. If $P (a, - 3)$ has image $(5, b)$ in the x-axis, then $a = 5$ and $b = 3$ .

Graph-paper questions (5 marks). Both 2024 Q3(iii) and 2025 Q6(i) were graph questions. The marks go for: the given scale (2 cm = 1 unit), correctly placed axes, every point plotted and labelled with its coordinates, the images found, the points joined in the stated order, and the figure named by its special property (in 2024 a square, not just a quadrilateral; in 2025 a decagon). The examiners reported candidates who swapped positive and negative axes, plotted $(3, 0)$ on the y-axis, reflected in the wrong axis, and left points or figures unnamed.

Properties kept by reflection. Lengths and angles do not change, so the image of a figure is congruent to it. This is why figures formed by a point and its images are often symmetric: rectangles, squares, rhombuses, isosceles trapeziums and regular-looking polygons.

Reflection in the Axes and the Origin: formulas to remember

Reflection in the x-axis: $M_{x} : (x, y) \to (x, - y)$; y changes sign.
Reflection in the y-axis: $M_{y} : (x, y) \to (- x, y)$; x changes sign.
Reflection in the origin: $M_{O} : (x, y) \to (- x, - y)$; Both change sign.
Combinations: $M_{y} (M_{x} (P)) = M_{x} (M_{y} (P)) = M_{O} (P)$; One axis then the other = the origin.

How to solve Reflection in the Axes and the Origin questions

Name the mirror, change the right sign, then plot and label everything.

1Identify the mirror: x-axis, y-axis or origin.
2Apply the rule: x-axis changes $y$ ; y-axis changes $x$ ; origin changes both.
3For a chain of reflections, apply them one at a time, writing each image.
4On graph paper, use the given scale, draw both axes with arrows, and mark negative values carefully.
5Plot and label every point with its coordinates, join in the stated order, and name the figure precisely.
6If asked for an area, use the figure's shape (rectangle, trapezium, triangle) or split it into simple parts.

Reflection in the Axes and the Origin 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.

Images in both axes and the origin

Suppose a kaleidoscope app uses the two axes as mirrors, so a dot drawn at $(2, 5)$ is copied by reflection in each axis and in the origin.

Where do the three copies appear?

Try it first, then show the solutionSolution

In the y-axis: $(- 2, 5)$ .
In the x-axis: $(2, - 5)$ .
In the origin: $(- 2, - 5)$ .

Answer: $(- 2, 5)$ , $(2, - 5)$ and $(- 2, - 5)$ : one in each quadrant.

Memory hook: Two mirrors, four copies, one per quadrant.

Distance between a point and its image

Imagine a mirror-image puzzle game in which a player at $P (- 4, 3)$ sees copies of itself reflected in the x-axis and in the origin.

How far is the player from each copy?

Try it first, then show the solutionSolution

Copy in the x-axis: $(- 4, - 3)$ ; distance $= 3 - (- 3) = 6$ .
Copy in the origin: $(4, - 3)$ ; distance $= 8^{2} + 6^{2} = 10$ , which is twice $O P = 5$ .

Answer: 6 units and 10 units.

Memory hook: The mirror is halfway: distance to the image is twice the distance to the mirror.

Tricks to remember Reflection in the Axes and the Origin

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

The axis keeps its own letter
Reflect in the x-axis: x is kept, y flips. Reflect in the y-axis: y is kept, x flips.
Use it when: Avoiding the axis mix-up reported in 2024 and 2025.
Origin = both flips
Through the origin, both signs change; it equals x-axis then y-axis.
Use it when: Reflection in the origin and combined reflections.
Label as you plot
Write the coordinates beside each point the moment you plot it.
Use it when: Getting the presentation marks in 5-mark graph questions.

Reflection in the Axes and the Origin: solved examples

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

Example 1: Image in the x-axis · 1 mark · MCQBoard level

The image of $(- 6, 2)$ under reflection in the x-axis is:
(a) $(6, 2)$
(b) $(- 6, - 2)$
(c) $(6, - 2)$
(d) $(2, - 6)$

Try it first, then show the solutionSolution

In the x-axis, $x$ stays and $y$ changes sign: $(- 6, 2) \to (- 6, - 2)$ .

Answer: (b) $(- 6, - 2)$

Example 2: All three images · 3 marks · Short answerBoard level

Write the images of $P (4, - 3)$ under reflection in (a) the x-axis (b) the y-axis (c) the origin.

Try it first, then show the solutionSolution

(a) $M_{x}$ : $(4, 3)$ .
(b) $M_{y}$ : $(- 4, - 3)$ .
(c) $M_{O}$ : $(- 4, 3)$ .

Answer: (a) $(4, 3)$ (b) $(- 4, - 3)$ (c) $(- 4, 3)$

Example 3: Finding unknowns from an image · 3 marks · Short answerHarder board level

$P (a, - 3)$ is reflected in the x-axis to $P^{'} (5, b)$ .
(a) Find $a$ and $b$ .
(b) Write the image of $P$ under reflection in the origin.

Try it first, then show the solutionSolution

(a) $M_{x} (a, - 3) = (a, 3)$ . Comparing with $(5, b)$ : $a = 5$ , $b = 3$ .
(b) $P (5, - 3) \to (- 5, 3)$ .

Answer: (a) $a = 5$ , $b = 3$ (b) $(- 5, 3)$

Example 4: A chain of reflections and the figure formed · 4 marks · Short answerHarder board level

$A (3, 1)$ is reflected in the y-axis to $B$ ; $B$ is reflected in the x-axis to $C$ ; $C$ is reflected in the y-axis to $D$ .
(a) Write the coordinates of $B$ , $C$ , $D$ .
(b) Name the figure $A B C D$ and find its area.

Try it first, then show the solutionSolution

(a) $B (- 3, 1)$ , $C (- 3, - 1)$ , $D (3, - 1)$ . Note $D$ is also the image of $A$ in the x-axis.
(b) $A B$ and $D C$ are horizontal (length 6); $B C$ and $A D$ are vertical (length 2). All angles are right angles, so $A B C D$ is a rectangle.
Area $= 6 \times 2 = 12$ square units.

Answer: (a) $B (- 3, 1)$ , $C (- 3, - 1)$ , $D (3, - 1)$ (b) Rectangle, 12 square units.

Example 5: Graph question: an isosceles trapezium · 5 marks · Long answerHOTS / competency

Use graph paper (2 cm = 1 unit).
(a) Plot $P (2, 4)$ and $Q (4, 1)$ .
(b) Reflect $P$ and $Q$ in the x-axis to $P^{'}$ and $Q^{'}$ ; write their coordinates.
(c) Join $P$ , $Q$ , $Q^{'}$ , $P^{'}$ and name the figure.
(d) Find its area.

Try it first, then show the solutionSolution

(b) $P^{'} (2, - 4)$ , $Q^{'} (4, - 1)$ .
(c) $P P^{'}$ (on $x = 2$ , length 8) and $Q Q^{'}$ (on $x = 4$ , length 2) are both vertical, so parallel. $P Q = 2^{2} + 3^{2} = 13 = P^{'} Q^{'}$ . One pair of parallel sides and equal non-parallel sides: an isosceles trapezium.
(d) Parallel sides 8 and 2, distance between them $4 - 2 = 2$ : area $= \frac{1}{2} (8 + 2) \times 2 = 10$ square units.

Answer: $P^{'} (2, - 4)$ , $Q^{'} (4, - 1)$ ; isosceles trapezium; area 10 square units.

Example 6: Case study: an octagonal logo · 5 marks · Case-basedHOTS / competency

A designer plots $A (1, 4)$ and $B (3, 2)$ and reflects them to build a symmetric logo.
(a) Reflect $A$ and $B$ in the x-axis, the y-axis and the origin, and write all six images.
(b) Joining $A$ , $B$ , the image of $B$ in the x-axis, the image of $A$ in the x-axis, then the origin images and the y-axis images in order, name the figure.
(c) Find its area.

Try it first, then show the solutionSolution

(a) x-axis: $(1, - 4)$ , $(3, - 2)$ . y-axis: $(- 1, 4)$ , $(- 3, 2)$ . Origin: $(- 1, - 4)$ , $(- 3, - 2)$ .
(b) Order: $(1, 4), (3, 2), (3, - 2), (1, - 4), (- 1, - 4), (- 3, - 2), (- 3, 2), (- 1, 4)$ : 8 vertices, an octagon.
(c) It fits in the rectangle from $x = - 3$ to $3$ and $y = - 4$ to $4$ , area $6 \times 8 = 48$ . Four corner triangles, each with legs 2 and 2, are cut off: $4 \times \frac{1}{2} \times 2 \times 2 = 8$ .
Area $= 48 - 8 = 40$ square units.

Answer: (b) An octagon (c) 40 square units

Where students lose marks in Reflection in the Axes and the Origin

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

Mistake: Reflecting in the wrong axis
In the x-axis the y-coordinate changes, and vice versa.
Wrong working
Reflect $B (2, 1)$ in the y-axis: $(2, - 1)$ .
Right working
$(- 2, 1)$
Do this: Say 'x-axis keeps x' or 'y-axis keeps y' before writing.
Usually costs: Reported in 2024 Q3(iii) and 2025 Q6(i); every later part goes wrong.
Mistake: Wrong scale or swapped axes
The graph must follow the stated scale with negative values left and down.
Wrong working
Plotting $C (3, 0)$ at $(0, 3)$ on the y-axis.
Right working
$(3, 0)$ lies on the x-axis, 3 units right of $O$ .
Do this: Write the scale in a corner and number both axes before plotting.
Usually costs: Reported in 2025 Q6(i); plotting marks.
Mistake: Not naming points or the figure
Marks are given for coordinates of every image and a precise name.
Wrong working
Points plotted but unlabelled; figure called 'a polygon'.
Right working
Each point labelled, e.g. $F (- 1, - 3)$ ; figure named 'decagon' or 'square'.
Do this: Finish with a sentence: 'The figure is a …'.
Usually costs: Reported in both years.

Exam tips

2025 Q6(i) (5 marks) reflected points in the x-axis, the origin and the y-axis and asked for the figure: a decagon. Count the vertices before naming.
Name the figure by its strongest property: square rather than rhombus or quadrilateral, isosceles trapezium rather than trapezium.
Use the scale given in the question (2 cm = 1 unit); a different scale loses marks.
List all coordinates in your answer as well as on the graph.

Practice questions on Reflection in the Axes and the Origin

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 image of $(0, - 5)$ under reflection in the origin is:
(a) $(0, - 5)$
(b) $(0, 5)$
(c) $(5, 0)$
(d) $(- 5, 0)$
Hint
Both signs change; $- 0 = 0$ .
Answer
(b) $(0, 5)$
Q2 · 2 marks · Short answerBoard level
Write the images of $(- 3, - 7)$ in the x-axis and in the y-axis.
Hint
x-axis changes $y$ ; y-axis changes $x$ .
Answer
$(- 3, 7)$ and $(3, - 7)$
Q3 · 3 marks · Short answerHarder board level
A point $A$ is reflected in the origin to $A^{'} (3, - 4)$ , and $A^{'}$ is then reflected in the x-axis to $A^{''}$ . Find $A$ and $A^{''}$ , and name the single reflection that maps $A$ to $A^{''}$ .
Hint
Undo the origin reflection to get $A$ .
Answer
$A (- 3, 4)$ , $A^{''} (3, 4)$ ; the single reflection is in the y-axis.
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): The point $(0, 4)$ is its own image under reflection in the y-axis.
Reason (R): A point on the y-axis has $x = 0$ , and $- 0 = 0$ .
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
$M_{y} (0, 4) = (- 0, 4)$ .
Answer
(c)
Q5 · 3 marks · Short answerHarder board level
Triangle $A B C$ has $A (1, 2)$ , $B (4, 2)$ , $C (1, 5)$ . Reflect it in the origin and write the coordinates of $A^{'}$ , $B^{'}$ , $C^{'}$ . What is the midpoint of $A A^{'}$ ?
Hint
Change both signs.
Answer
$A^{'} (- 1, - 2)$ , $B^{'} (- 4, - 2)$ , $C^{'} (- 1, - 5)$ ; the midpoint of $A A^{'}$ is the origin $(0, 0)$ .
Q6 · 3 marks · Short answerHarder board level
Show that reflecting $P (x, y)$ in the x-axis and then reflecting the image in the origin is the same as reflecting $P$ in the y-axis. Use this to find the final image of $(3, 7)$ .
Hint
$(x, y) \to (x, - y) \to (- x, y)$ .
Answer
$M_{O} (M_{x} (x, y)) = M_{O} (x, - y) = (- x, y) = M_{y} (x, y)$ . For $(3, 7)$ : $(- 3, 7)$ .
Q7 · 4 marks · Short answerHOTS / competency
$A (3, 2)$ is reflected in the x-axis to $B$ and in the y-axis to $C$ . Find the area of triangle $A B C$ and the angle at $A$ .
Hint
$A B$ is vertical and $A C$ is horizontal.
Answer
$B (3, - 2)$ , $C (- 3, 2)$ . $A B = 4$ , $A C = 6$ and $∠ A = 90°$ ; area $= \frac{1}{2} \times 4 \times 6 = 12$ square units.

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.Image of $(5, - 1)$ in the y-axis?Show answer
$(- 5, - 1)$
2.Image of $(- 2, - 8)$ in the origin?Show answer
$(2, 8)$
3.Which two reflections together make reflection in the origin?Show answer
Reflection in the x-axis and reflection in the y-axis (either order).
4.What does the mirror line do to the segment joining a point and its image?Show answer
It is the perpendicular bisector of that segment.
5.Image of $(7, 0)$ in the x-axis?Show answer
$(7, 0)$ itself.

Reflection in the Axes and the Origin: frequently asked questions

What are the rules for reflection in the x-axis, y-axis and origin?

x-axis: (x, y) → (x, −y). y-axis: (x, y) → (−x, y). Origin: (x, y) → (−x, −y). The coordinate measuring distance from the mirror changes sign.

Is reflection in the origin the same as two reflections?

Yes. Reflecting in the x-axis and then the y-axis (in either order) gives (−x, −y), the same as reflecting in the origin.

How are graph-paper reflection questions marked in ICSE?

Marks go for the correct scale and axes, plotting and labelling every point with coordinates, correct images, joining the points in order, and naming the figure precisely.