MathsICSE Class 10 · Chapter 10

Reflection ICSE Class 10: Rules, Invariant Points and Graph Questions

Updated 14 October 2026 · Fact-checked

Reflecting a point in a line puts its image the same distance on the other side, along a perpendicular. In ICSE Class 10 you reflect points in the x-axis, the y-axis, the origin and lines x = a and y = a, find invariant points, and plot figures on graph paper.

What Reflection is about

A reflection in a line works like a mirror: the image of a point is as far behind the mirror line as the point is in front of it, and the line joining a point to its image is perpendicular to the mirror line and is bisected by it.

The CISCE 2027 syllabus covers: (i) reflection of a point in a line: $x = 0$ (the y-axis), $y = 0$ (the x-axis), $x = a$ , $y = a$ , and the origin; (ii) reflection of a point in the origin; (iii) invariant points.

In the 2027 specimen paper, the 5-mark part of Question 3 asks you to read points off a given graph and reflect them, and the 2024 and 2025 board papers each had a 5-mark graph-paper reflection question. Those questions are easy marks if you follow the drill: choose the scale given, label the axes correctly, plot every point, write the coordinates of every image, join in the stated order and name the figure precisely. The examiners' reports for both years list marks lost on each of these steps, and on reflecting in the wrong axis.

Weightage: CISCE publishes no chapter or unit weightage for ICSE Mathematics. Reflection is in the 2. Algebra (vii) Co-ordinate Geometry (a) Reflection part of the 2027 syllabus and can be asked in both Section A and Section B of the 80-mark paper.Source: CISCE ICSE Mathematics syllabus, examination year 2027

Key concepts

Reflection in the x-axis and the y-axis

In the x-axis ( $y = 0$ ): $(x, y) \to (x, - y)$ . The x-coordinate stays; the y-coordinate changes sign.
In the y-axis ( $x = 0$ ): $(x, y) \to (- x, y)$ . The y-coordinate stays; the x-coordinate changes sign.

A quick check: the coordinate that changes sign is the one that measures distance from the mirror. Reflecting in the x-axis flips a point up or down, so its height (y) changes sign.

Notation: $M_{x}$ means reflection in the x-axis, $M_{y}$ in the y-axis, $M_{O}$ in the origin.

Worked example

Write the images of $P (3, - 5)$ under $M_{x}$ , $M_{y}$ and $M_{O}$ .

Try it first, then show the solutionSolution

$M_{x}$ : $(3, 5)$ .
$M_{y}$ : $(- 3, - 5)$ .
$M_{O}$ : $(- 3, 5)$ .

Answer: (3, 5), (−3, −5), (−3, 5)

Reflection in the origin

Reflection in the origin sends $(x, y) \to (- x, - y)$ : both coordinates change sign. The origin is the midpoint of a point and its image.

It has the same effect as reflecting in the x-axis and then in the y-axis (in either order): $(x, y) \to (x, - y) \to (- x, - y)$ .

Worked example

Reflect $P (2, 5)$ in the x-axis and then reflect the image in the y-axis. Name the single reflection that does the same.

Try it first, then show the solutionSolution

$M_{x}$ : $(2, - 5)$ .
$M_{y}$ : $(- 2, - 5)$ .
$(2, 5) \to (- 2, - 5)$ is reflection in the origin.

Answer: (−2, −5); reflection in the origin

Reflection in the lines x = a and y = a

In the vertical line $x = a$ : $(x, y) \to (2 a - x, y)$ .
In the horizontal line $y = a$ : $(x, y) \to (x, 2 a - y)$ .

Why? The mirror line is halfway between the point and its image, so $a = \frac{x + x ^{'}}{2}$ , giving $x^{'} = 2 a - x$ . You can also count squares: if $P$ is 3 units to the right of $x = 2$ , its image is 3 units to the left.

Worked example

Reflect $P (5, 1)$ in the line $x = 2$ , and $Q (4, 2)$ in the line $y = - 3$ .

Try it first, then show the solutionSolution

$P^{'}$ : $(2 \times 2 - 5, 1) = (- 1, 1)$ .
$Q^{'}$ : $(4, 2 (- 3) - 2) = (4, - 8)$ .

Answer: P′(−1, 1), Q′(4, −8)

Invariant points

A point is invariant under a reflection if its image is the point itself. For a reflection in a line, the invariant points are exactly the points on the mirror line. For a reflection in the origin, the only invariant point is the origin itself.

So: invariant under $M_{x}$ means $y = 0$ ; under $M_{y}$ , $x = 0$ ; under reflection in $y = 3$ , the y-coordinate must be 3.

Worked example

Which of $(2, 0)$ , $(0, 3)$ , $(- 4, 0)$ are invariant under reflection in the x-axis? For what value of $b$ is $(7, b)$ invariant under reflection in $y = 3$ ?

Try it first, then show the solutionSolution

Invariant under $M_{x}$ needs $y = 0$ : $(2, 0)$ and $(- 4, 0)$ .
Invariant under reflection in $y = 3$ needs $b = 3$ .

Answer: (2, 0) and (−4, 0); b = 3

Finding the mirror line from a point and its image

The mirror line is the perpendicular bisector of the segment joining a point and its image.

If the x-coordinates are equal (the segment is vertical), the mirror is horizontal: $y = \frac{y _{1} + y _{2}}{2}$ .
If the y-coordinates are equal, the mirror is vertical: $x = \frac{x _{1} + x _{2}}{2}$ .
If both coordinates change sign, it is reflection in the origin.

Worked example

Find the mirror line if (a) $A (4, 1) \to A^{'} (4, 7)$ , (b) $B (- 3, 2) \to B^{'} (5, 2)$ .

Try it first, then show the solutionSolution

(a) Same x, so the line is $y = \frac{1 + 7}{2} = 4$ .
(b) Same y, so the line is $x = \frac{- 3 + 5}{2} = 1$ .

Answer: (a) y = 4 (b) x = 1

Graph-paper questions and naming the figure

Drill for every graph question:
1. Use the scale given (e.g. 2 cm = 1 unit) on both axes; label $x$ , $x^{'}$ , $y$ , $y^{'}$ and $O$ ; mark positive and negative sides correctly.
2. Plot and label each given point.
3. Find each image, plot it and write its coordinates next to it.
4. Join the points in the order stated.
5. Name the figure precisely (square, rectangle, isosceles trapezium, kite, hexagon, decagon…), using properties you can see: equal sides, right angles, parallel sides, symmetry.
6. If asked, find the area, using lengths along the grid.

Worked example

$A (2, 3)$ is reflected in the x-axis to $B$ , in the y-axis to $D$ and in the origin to $C$ . Name the figure $A B C D$ and find its area.

Try it first, then show the solutionSolution

$B (2, - 3)$ , $D (- 2, 3)$ , $C (- 2, - 3)$ .
$A B$ and $D C$ are vertical (length 6), $A D$ and $B C$ horizontal (length 4): a rectangle.
Area $= 6 \times 4 = 24$ square units.

Answer: Rectangle; 24 square units

Maths in real life: Reflection

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 as a fold

Suppose you draw half a rangoli design on graph paper to the right of the y-axis, and fold the sheet along the y-axis so that the ink prints the other half.

Where does the dot at $(3, 2)$ print? Where does the dot at $(0, 5)$ print?

Try it first, then show the solutionSolution

Folding along the y-axis is reflection in $x = 0$ : $(x, y) \to (- x, y)$ .
$(3, 2) \to (- 3, 2)$ .
$(0, 5)$ is on the fold, so it stays at $(0, 5)$ : an invariant point.

Answer: (−3, 2); (0, 5) stays where it is.

Memory hook: The fold line is the mirror; dots on the fold do not move.

Reflection in a horizontal line y = a

Imagine a map of a park drawn on a grid, with the edge of a still pond along the line $y = 1$ . A tree stands at $(4, 5)$ .

Where does the tree's reflection appear on the map?

Try it first, then show the solutionSolution

Reflection in $y = 1$ : $(x, y) \to (x, 2 - y)$ .
$(4, 5) \to (4, - 3)$ : 4 units above the edge becomes 4 units below.

Answer: (4, −3)

Memory hook: As far behind the mirror as in front of it.

Symmetric design about x = a

Suppose you design a kite on a grid so that it is symmetric about the line $x = 2$ , and one corner is at $(5, 3)$ .

Where is the matching corner?

Try it first, then show the solutionSolution

Reflection in $x = 2$ : $(x, y) \to (4 - x, y)$ .
$(5, 3) \to (- 1, 3)$ .

Answer: (−1, 3)

Memory hook: Symmetry line = mirror line.

Topics in this chapter

Reflection formula sheet

Reflection in the x-axis: $M_{x} : (x, y) \to (x, - y)$; Invariant points: y = 0.
Reflection in the y-axis: $M_{y} : (x, y) \to (- x, y)$; Invariant points: x = 0.
Reflection in the origin: $M_{O} : (x, y) \to (- x, - y)$; Only the origin is invariant.
Reflection in x = a: $(x, y) \to (2 a - x, y)$; Vertical mirror.
Reflection in y = a: $(x, y) \to (x, 2 a - y)$; Horizontal mirror.
Mirror line from point and image: $x = \frac{x _{1} + x _{2}}{2} or y = \frac{y _{1} + y _{2}}{2}$; The mirror bisects the segment at right angles.
Two reflections: $M_{y} (M_{x} (P)) = M_{O} (P)$; x-axis then y-axis equals the origin.

How the board asks Reflection

The question patterns to expect, and the method that works for each.

Question type	How it is asked	Method
MCQ: image of a point	Section A, Question 1: the image of a point in an axis, the origin or a line like y = 2.	Apply the rule; check that the mirror bisects the segment.
Invariant points	Which points are invariant, or find k so that (k, 4) is invariant.	Invariant means on the mirror line.
Find the point, the mirror or the unknown	A point and its image are given with unknowns; find them or the mirror line.	Use midpoint on the mirror line; the coordinate parallel to the mirror stays the same.
Graph-paper question (5 marks)	Plot points, reflect them in stated lines, write coordinates, join and name the figure, sometimes find its area.	Follow the drill: scale, axes, plot, coordinates, join in order, name precisely.
Combined reflections	Reflect in one line then another; name the single transformation or describe the result.	Apply one rule at a time, then compare the original and final points.

Tricks to remember Reflection

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

Mirror keeps its own coordinate
x-axis mirror keeps x; y-axis mirror keeps y. The other coordinate flips sign.
Use it when: M_x and M_y without mixing them up.
Origin flips both
Through the origin, both signs flip: it is a half-turn.
Use it when: Reflection in the origin.
Twice the line minus the point
In x = a: new x = 2a − x. In y = a: new y = 2a − y.
Use it when: Lines other than the axes.
On the mirror, nothing moves
Invariant points lie on the mirror line.
Use it when: Invariant-point questions.

Reflection: board-level solved questions

Ordered from board level to HOTS. Cover the solution, write your own, then compare step by step.

Example 1: Image in the origin · 1 mark · MCQBoard level

The image of $(- 4, 7)$ under reflection in the origin is:
(a) $(4, 7)$
(b) $(- 4, - 7)$
(c) $(4, - 7)$
(d) $(7, - 4)$

Try it first, then show the solutionSolution

Both signs change: $(4, - 7)$ .

Answer: (c)

Example 2: An invariant point · 1 mark · Assertion-reasonBoard level

Assertion (A): The point $(5, 0)$ is invariant under reflection in the x-axis.
Reason (R): Points lying on the mirror line are invariant under reflection in that line.
(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).

Try it first, then show the solutionSolution

$(5, 0) \to (5, 0)$ under $M_{x}$ , and it lies on the x-axis, the mirror.

Answer: (c)

Example 3: Work back, then reflect in y = 2 · 4 marks · Short answerHarder board level

A point $P$ is reflected in the x-axis to $P^{'} (5, - 3)$ . (a) Write the coordinates of $P$ . (b) $P$ is reflected in the line $y = 2$ to $P^{''}$ . Find $P^{''}$ . (c) Find the distance $P P^{''}$ .

Try it first, then show the solutionSolution

(a) Reflecting back in the x-axis: $P (5, 3)$ .
(b) $P^{''} = (5, 4 - 3) = (5, 1)$ .
(c) Same x, so $P P^{''} = 3 - 1 = 2$ units.

Answer: (a) (5, 3) (b) (5, 1) (c) 2 units

Example 4: Find the mirror and the unknown · 3 marks · Short answerHarder board level

The point $(- 2, k)$ is reflected in the line $x = a$ and its image is $(6, 3)$ . Find $a$ and $k$ .

Try it first, then show the solutionSolution

Reflection in $x = a$ does not change y, so $k = 3$ .
$a$ is the midpoint of the x-coordinates: $a = \frac{- 2 + 6}{2} = 2$ .

Answer: a = 2, k = 3

Example 5: Two reflections, one result · 4 marks · Short answerHarder board level

$A (3, - 2)$ is reflected in the x-axis to $A^{'}$ , and $A^{'}$ is reflected in the origin to $A^{''}$ . Find $A^{'}$ and $A^{''}$ , and name the single reflection that maps $A$ to $A^{''}$ .

Try it first, then show the solutionSolution

$A^{'} = (3, 2)$ .
$A^{''} = (- 3, - 2)$ .
$A (3, - 2) \to (- 3, - 2)$ : only the x-coordinate changed sign, so this is reflection in the y-axis.

Answer: A′(3, 2), A″(−3, −2); reflection in the y-axis

Example 6: Reflection in x = 1, then in the x-axis · 3 marks · Long answerBoard level

$A (- 3, 2)$ is reflected in the line $x = 1$ to $A^{'}$ , and $A^{'}$ is reflected in the x-axis to $A^{''}$ . (a) Write the coordinates of $A^{'}$ and $A^{''}$ . (b) Write the coordinates of the point that is invariant under reflection in $x = 1$ and also lies on the x-axis.

Try it first, then show the solutionSolution

Reflection in $x = 1$ : $(x, y) \to (2 - x, y)$ , so $A^{'} = (2 + 3, 2) = (5, 2)$ .
Reflection in the x-axis: $A^{''} = (5, - 2)$ .
(b) Invariant points of the reflection lie on $x = 1$ ; the one on the x-axis is $(1, 0)$ .

Answer: (a) A′(5, 2), A″(5, −2) (b) (1, 0)

Example 7: Graph question with a line y = 3 · 5 marks · Long answerHOTS / competency

Use graph paper (2 cm = 1 unit). (a) Plot $A (2, 1)$ and $B (5, 1)$ . (b) Reflect $A$ and $B$ in the line $y = 3$ to $A^{'}$ and $B^{'}$ ; write their coordinates. (c) Name the figure $A B B^{'} A^{'}$ and find its area. (d) Which points of the boundary of $A B B^{'} A^{'}$ are invariant under reflection in $y = 3$ ? Explain.

Try it first, then show the solutionSolution

(b) $A^{'} (2, 5)$ , $B^{'} (5, 5)$ .
(c) $A B = A^{'} B^{'} = 3$ (horizontal), $A A^{'} = B B^{'} = 4$ (vertical): a rectangle. Area $= 3 \times 4 = 12$ square units.
(d) Invariant points must lie on $y = 3$ . The boundary meets $y = 3$ only on the sides $A A^{'}$ and $B B^{'}$ , at $(2, 3)$ and $(5, 3)$ .

Answer: A′(2, 5), B′(5, 5); rectangle of area 12; (2, 3) and (5, 3)

Example 8: Mirror line from a point and its image · 4 marks · Long answerHarder board level

$P (- 1, 4)$ is reflected in a line $L$ to $P^{'} (5, 4)$ . (a) Find the equation of $L$ . (b) Find the image of $Q (0, - 3)$ in $L$ .

Try it first, then show the solutionSolution

(a) Same y, so $L$ is vertical: $x = \frac{- 1 + 5}{2} = 2$ .
(b) $Q^{'} = (2 \times 2 - 0, - 3) = (4, - 3)$ .

Answer: (a) x = 2 (b) (4, −3)

Case-study practice (Section B style)

Read the situation, then answer every part. In the board paper this is a 10-mark question split into 3 + 3 + 4 marks.

Designing a badge on graph paper

Neha designs a school badge on graph paper (2 cm = 1 unit). She plots $P (2, 3)$ and $Q (5, 1)$ and uses reflections to complete a symmetric shape.

(i) · 3 marks
Reflect $P$ and $Q$ in the x-axis to $P^{'}$ and $Q^{'}$ . Write their coordinates.
Answer
Answer: P′(2, −3), Q′(5, −1)
$M_{x}$ changes the sign of y: $P^{'} (2, - 3)$ , $Q^{'} (5, - 1)$ .
(ii) · 3 marks
Join $P$ , $Q$ , $Q^{'}$ , $P^{'}$ in order. Name the figure and give a reason.
Answer
Answer: An isosceles trapezium
$P P^{'}$ (x = 2, length 6) and $Q Q^{'}$ (x = 5, length 2) are both vertical, so they are parallel and unequal: a trapezium. $P Q = P^{'} Q^{'} = 9 + 4 = 13$ , so it is isosceles (the x-axis is a line of symmetry).
(iii) · 4 marks
Find the area of $P Q Q^{'} P^{'}$ , and state the invariant points of its boundary under reflection in the x-axis.
Answer
Answer: 12 square units; (2, 0) and (5, 0)
Area $= \frac{1}{2} (6 + 2) \times 3 = 12$ square units (the parallel sides are 3 units apart). The boundary crosses the x-axis at $(2, 0)$ on $P P^{'}$ and $(5, 0)$ on $Q Q^{'}$ ; these are the invariant points.

Finding a hidden mirror

In a puzzle, a point $P (3, 5)$ is reflected in an unknown line $L$ to $P^{'} (3, - 1)$ .

(i) · 3 marks
Find the equation of $L$ .
Answer
Answer: y = 2
Same x, so $L$ is horizontal through the midpoint: $y = \frac{5 + ( - 1 )}{2} = 2$ .
(ii) · 3 marks
Find the image of $Q (- 2, 4)$ in $L$ .
Answer
Answer: (−2, 0)
$(x, 2 \times 2 - y) = (- 2, 0)$ .
(iii) · 4 marks
Which points are invariant under reflection in $L$ ? Find the invariant point that lies on the y-axis, and explain why $Q$ 's image lies on the x-axis.
Answer
Answer: All points with y = 2; (0, 2)
Points on $L$ ( $y = 2$ ) are invariant; on the y-axis this is $(0, 2)$ . $Q$ is 2 units above $L$ , so its image is 2 units below $L$ , at $y = 0$ , which is the x-axis.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Write the images of $A (- 3, 4)$ under reflection in the x-axis, the y-axis and the origin. (3 marks)

Step	Marks
$M_{x}$ : $(- 3, - 4)$	1
$M_{y}$ : $(3, 4)$	1
$M_{O}$ : $(3, - 4)$	1

$P (- 1, 2)$ is reflected in a line to $P^{'} (- 1, 8)$ . Find the line, and the image of $R (4, 1)$ in the same line. (4 marks)

Step	Marks
x-coordinates equal, so the mirror is horizontal	1
$y = \frac{2 + 8}{2} = 5$	1
Image of R: $(4, 2 \times 5 - 1)$	1
$= (4, 9)$	1

On graph paper (2 cm = 1 unit), plot $A (3, 2)$ and $B (1, 4)$ . Reflect them in the x-axis to $A^{'}$ and $B^{'}$ . Join $A$ , $B$ , $B^{'}$ , $A^{'}$ in order, name the figure and find its area. (5 marks)

Step	Marks
Correct scale, labelled axes, A and B plotted	1
$A^{'} (3, - 2)$ , $B^{'} (1, - 4)$ plotted with coordinates written	1
Points joined in the order A, B, B′, A′	1
Isosceles trapezium ( $A A^{'} ∥ B B^{'}$ , $A B = A^{'} B^{'}$ )	1
Area $= \frac{1}{2} (4 + 8) \times 2 = 12$ square units	1

Where students lose marks in Reflection

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

Mistake: Reflecting in the wrong axis
CISCE examiners' comments, 2024 and 2025: candidates reflected in the x-axis when the y-axis was asked, and vice versa.
Wrong working
B(2, 1) reflected in the y-axis: B′(2, −1)
Right working
B′(−2, 1): the y-axis mirror keeps y and changes the sign of x
Do this: Say the rule aloud: 'the y-axis keeps y'. Check on the graph that the image is on the other side of the correct axis.
Usually costs: 1 to 2 marks in the graph question.
Mistake: Reflecting in an axis instead of a given line
CISCE examiners' comments, 2024: asked to reflect A in the line BB′, candidates reflected it in the x-axis and got (0, −3).
Do this: Find the equation of the given line first (e.g. y = 1), then use $(x, 2 a - y)$ .
Usually costs: The coordinates mark and the figure that depends on it.
Mistake: Wrong scale, swapped axes, unlabelled coordinates
CISCE examiners' comments, 2024 and 2025: wrong scale chosen, positive and negative axes interchanged, points plotted wrongly (e.g. (3, 0) put on the y-axis), and coordinates of images not written.
Do this: Use exactly the scale given; mark O, x, x′, y, y′; write every point's coordinates beside it.
Usually costs: 1 mark each.
Mistake: Not naming the figure, or naming it loosely
CISCE examiners' comments, 2024 and 2025: figures were named 'quadrilateral', 'rhombus' or 'kite' when they were squares, and some candidates did not name the decagon at all.
Do this: Check sides and angles on the grid (equal lengths, right angles, parallel sides) and give the most precise name.
Usually costs: 1 mark.
Mistake: Not joining in the order stated
CISCE examiners' comments, 2025: candidates did not join all the points in order, so the closed figure was wrong.
Do this: Join exactly in the order the question lists, and close the figure back to the first point.
Usually costs: 1 mark.

Practice questions on Reflection

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 $(3, 0)$ under reflection in the y-axis is:
(a) $(3, 0)$
(b) $(- 3, 0)$
(c) $(0, 3)$
(d) $(0, - 3)$
Hint
Change the sign of x.
Answer
(b)
Q2 · 1 mark · MCQBoard level
The only point that is invariant under reflection in the origin is:
(a) $(1, 1)$
(b) $(0, 1)$
(c) $(0, 0)$
(d) every point on the x-axis
Hint
Both coordinates must equal their negatives.
Answer
(c)
Q3 · 1 mark · MCQHarder board level
The image of $(5, - 2)$ in the line $y = 1$ is:
(a) $(5, 4)$
(b) $(- 3, - 2)$
(c) $(5, 3)$
(d) $(5, 0)$
Hint
(x, 2a − y).
Answer
(a): $2 - (- 2) = 4$ .
Q4 · 1 mark · Assertion-reasonBoard level
Assertion (A): Reflecting a point in the x-axis and then in the y-axis gives the same result as reflecting it in the origin.
Reason (R): $(x, y) \to (x, - y) \to (- x, - y)$ .
(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
Follow the coordinates.
Answer
(c)
Q5 · 4 marks · Short answerHarder board level
$A (2, 3)$ is reflected in the line $x = 4$ to $A^{'}$ , and $A^{'}$ is reflected in the line $y = - 1$ to $A^{''}$ . Find $A^{'}$ , $A^{''}$ and the length $A A^{''}$ correct to two decimal places.
Hint
Apply one rule at a time.
Answer
$A^{'} (6, 3)$ ; $A^{''} (6, - 5)$ . $A A^{''} = 4^{2} + 8^{2} = 80 = 8.94$ units.
Q6 · 4 marks · Short answerHarder board level
A point $P$ is reflected in the y-axis to $(- 3, 7)$ . (a) Find $P$ . (b) Find the image of $P$ in the origin. (c) Find the line $x = k$ in which $P$ must be reflected to give $(- 5, 7)$ .
Hint
For (c), k is the midpoint of the x-coordinates.
Answer
(a) $P (3, 7)$ (b) $(- 3, - 7)$ (c) $k = \frac{3 + ( - 5 )}{2} = - 1$ , so $x = - 1$ .
Q7 · 3 marks · Long answerBoard level
The point $P (4, k)$ is reflected in the y-axis to $Q (m, - 3)$ . (a) Find $m$ and $k$ . (b) Write the image of $P$ under reflection in the origin. (c) Name the single reflection that maps $Q$ to that image.
Hint
Reflection in the y-axis keeps y and changes the sign of x.
Answer
(a) $m = - 4$ , $k = - 3$ , so $P (4, - 3)$ and $Q (- 4, - 3)$ . (b) $(- 4, 3)$ . (c) $Q (- 4, - 3) \to (- 4, 3)$ : only y changes sign, so it is reflection in the x-axis.
Q8 · 4 marks · Long answerHarder board level
Triangle $A B C$ has $A (1, 1)$ , $B (4, 1)$ , $C (4, 3)$ . (a) Reflect it in the y-axis to $A^{'} B^{'} C^{'}$ and write the coordinates. (b) Name the figure $B C C^{'} B^{'}$ and find its area.
Hint
B and B′ are 8 units apart.
Answer
(a) $A^{'} (- 1, 1)$ , $B^{'} (- 4, 1)$ , $C^{'} (- 4, 3)$ . (b) $B B^{'} = C C^{'} = 8$ (horizontal) and $B C = B^{'} C^{'} = 2$ (vertical): a rectangle of area 16 square units.
Q9 · 4 marks · Long answerHOTS / competency
A point $P (x, y)$ is reflected in the x-axis to $P^{'}$ , and $P^{'}$ is reflected in the line $y = 3$ to $P^{''}$ . Show that $P^{''}$ is always 6 units directly above $P$ . Check with $P (7, - 4)$ .
Hint
Track y.
Answer
$P^{'} (x, - y)$ ; $P^{''} (x, 6 - (- y)) = (x, y + 6)$ : same x and y increased by 6, so 6 units above. Check: $(7, - 4) \to (7, 4) \to (7, 2)$ , which is $(7, - 4 + 6)$ .
Q10 · 4 marks · Long answerHarder board level
$A (3, 4)$ and $B (- 1, 2)$ are reflected in the origin to $A^{'}$ and $B^{'}$ . Find $A^{'}$ and $B^{'}$ and show that $A B A^{'} B^{'}$ is a parallelogram.
Hint
Diagonals of a parallelogram bisect each other.
Answer
$A^{'} (- 3, - 4)$ , $B^{'} (1, - 2)$ . The origin is the midpoint of both $A A^{'}$ and $B B^{'}$ , so the diagonals of $A B A^{'} B^{'}$ bisect each other: it is a parallelogram.
Q11 · 1 mark · MCQHOTS / competency
Which point is invariant under reflection in both the line $x = 2$ and the line $y = - 1$ ?
(a) $(2, 1)$
(b) $(- 1, 2)$
(c) $(2, - 1)$
(d) $(0, 0)$
Hint
It must lie on both lines.
Answer
(c)
Q12 · 4 marks · Short answerHOTS / competency
$P (- 2, 3)$ is reflected in the line $x = a$ to $P^{'} (6, 3)$ , and $P^{'}$ is reflected in the line $y = b$ to $P^{''} (6, - 5)$ . Find $a$ and $b$ , and name the single transformation that maps $P$ to $P^{''}$ if $a$ and $b$ were both 0.
Hint
Midpoints give a and b.
Answer
$a = \frac{- 2 + 6}{2} = 2$ ; $b = \frac{3 + ( - 5 )}{2} = - 1$ . With $a = b = 0$ the two reflections are in the y-axis and the x-axis, which together give reflection in the origin.

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 (x, y) in the x-axis?Show answer
(x, −y)
2.Image of (x, y) in the y-axis?Show answer
(−x, y)
3.Image of (x, y) in the origin?Show answer
(−x, −y)
4.Image of (x, y) in the line x = a?Show answer
(2a − x, y)
5.Image of (x, y) in the line y = a?Show answer
(x, 2a − y)
6.What are invariant points?Show answer
Points whose image is themselves: the points on the mirror line.
7.Which single reflection equals M_x followed by M_y?Show answer
Reflection in the origin.
8.P(1, 6) has image (1, 0). What is the mirror line?Show answer
y = 3

Best order to study it

Learn M_x, M_y and M_O with many quick examples.
Learn reflection in x = a and y = a.
Practise invariant points and finding the mirror line.
Practise combined reflections.
Do full graph-paper questions with naming and area.

Quick revision checklist

The night before the exam: can you do each of these without looking?

Can you reflect a point in each axis and in the origin without hesitation?
Can you reflect in lines like x = 2 or y = −3?
Can you find invariant points?
Can you find the mirror line from a point and its image?
Can you name the single reflection equal to two reflections?
Do you write the coordinates of every image on the graph?
Can you name the figure precisely and find its area?

Reflection: frequently asked questions

Are reflections in lines like y = x in the ICSE 2027 syllabus?

The syllabus lists reflection in x = 0, y = 0, x = a, y = a and the origin. Practise those.

What scale should I use in a reflection graph question?

Use the scale given in the question, usually 2 cm = 1 unit on both axes. Examiners deduct marks for a different scale.

What is an invariant point?

A point that is its own image. Under reflection in a line, these are the points on that line; under reflection in the origin, only the origin.

Is reflection in the origin really a reflection?

In ICSE it is called reflection in the origin: each point goes to the point on the other side of the origin at the same distance, so the origin is the midpoint of a point and its image.

How do I name the figure after joining the points?

Look at side lengths along the grid, right angles and parallel sides. Use the most precise name: a square is better than a rhombus or a quadrilateral.

Based on: CISCE ICSE 2027 Mathematics syllabus; CISCE ICSE 2027 Mathematics specimen question paper and marking scheme; CISCE Analysis of Pupil Performance, ICSE Mathematics, 2025; CISCE Analysis of Pupil Performance, ICSE Mathematics, 2024