LIWA

Reflection

ICSE Class 10 · Mathematics · Chapter 12

What this chapter covers

Key points

Questions answered

Define the image of a point in a line.
The image of a point P in a line l is the point P' such that l is the perpendicular bisector of PP'.
Write the images of (7, -4) under reflection in (a) the x-axis, (b) the y-axis, (c) the origin.
(a) (7, 4) (b) (-7, -4) (c) (-7, 4).
What is an invariant point? Name the invariant points under reflection in the line y = -2.
An invariant point is a point that is its own image under the transformation. Under reflection in y = -2, every point (x, -2) on that line is invariant.
Find the image of (-3, 6) in the line x = 1.
(2 × 1 - (-3), 6) = (5, 6).
Find the image of (4, -5) in the line y = 2.
(4, 2 × 2 - (-5)) = (4, 9).
The point P(-4, b) is reflected in the x-axis to P'(a, 3). Find a and b.
Mx(-4, b) = (-4, -b), so a = -4 and -b = 3, giving b = -3.
A point (2, -1) is reflected to (2, 7). Name the line of reflection.
The abscissa is unchanged, so the line is horizontal; it is halfway between -1 and 7, so it is y = 3.
A point is reflected in the y-axis and then in the x-axis. Name the single transformation that has the same effect.
Reflection in the origin, since both coordinates change sign.

Past board exam questions

ICSE 2026 sample paper · 4 marks

Use a graph sheet for this question. Take 1 cm = 1 unit along both the x and y axis. Plot ABCDE, where A (4, 0), B (4, 2), C (2, 2), D (2,4) and E (0,4). (a) Reflect the points A, B, C and D on the y-axis and name them as F, G, H and I respectively. (b) Join the points A, B, C, D, E, I, H, G and F in order. Reflect the figure ABCDEIHGF on the x-axis and name it as AMNPQRSTF. (c) Give the geometrical name of the closed figure AEFQ.

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(a) Reflection in the y-axis: (x, y) → (−x, y). F(−4, 0), G(−4, 2), H(−2, 2), I(−2, 4). (b) Reflection in the x-axis: (x, y) → (x, −y). A(4, 0) and F(−4, 0) lie on the x-axis and stay fixed; B → M(4, −2), C → N(2, −2), D → P(2, −4), E → Q(0, −4), I → R(−2, −4), H → S(−2, −2), G → T(−4, −2). (c) A(4, 0), E(0, 4), F(−4, 0), Q(0, −4): all four sides equal 4√(2) and the diagonals AF and EQ are equal (8 units) and perpendicular, so AEFQ is a square.

ICSE 2026 board paper · 4 marks

Use graph sheet for this question. Take 2 cm = 1 unit along both x and y axis. Graphically represent parallelogram OABC, where O(0, 0), A(2, 3), B(5, 3) and C(3, 0). Reflect OABC: (a) on the x-axis and name its image as ODEC. (b) through the origin and name its image as OIJH. (c) on the y-axis and name its image as OFGH.

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Plot O(0, 0), A(2, 3), B(5, 3), C(3, 0) and join to form parallelogram OABC. (a) Reflection in the x-axis: (x, y) → (x, -y). D = A′ = (2, −3), E = B′ = (5, −3), C(3, 0) is invariant. ODEC: O(0, 0), D(2, −3), E(5, −3), C(3, 0). (b) Reflection through the origin: (x, y) → (-x, -y). I = (−2, −3), J = (−5, −3), H = (−3, 0). OIJH: O(0, 0), I(−2, −3), J(−5, −3), H(−3, 0). (c) Reflection in the y-axis: (x, y) → (-x, y). F = (−2, 3), G = (−5, 3), H = (−3, 0). OFGH: O(0, 0), F(−2, 3), G(−5, 3), H(−3, 0). Draw all four parallelograms on the same graph.

ICSE 2024 board paper · 5 marks

Use graph sheet for this question. Take 2 cm = 1 unit along the axes. (a) Plot A(0, 3), B(2, 1) and C(4, -1). (b) Reflect point B and C in y-axis and name their images as B' and C' respectively. Plot and write coordinates of the points B' and C'. (c) Reflect point A in the line BB' and name its images as A'. (d) Plot and write coordinates of point A'. (e) Join the points ABA'B' and give the geometrical name of the closed figure so formed.

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(a) A(0, 3), B(2, 1) and C(4, −1) are plotted. (b) Reflection in the y-axis: (x, y) → (-x, y). B' = (−2, 1), C' = (−4, −1). (c) BB' is the line y = 1. A(0, 3) is 2 units above it, so its image is 2 units below it. (d) A' = (0, −1). (e) In ABA'B', the diagonals AA' (length 4) and BB' (length 4) are equal, bisect each other at (0, 1) and are perpendicular. So ABA'B' is a square.

ICSE 2024 sample paper · 4 marks

Use graph sheet for this question. Take 2 cm = 1 unit along the axes. (a) Plot A (1, 2), B(1, 1)and C (2, 1) (b) Reflect A, B and C about y-axis and name them as Aʹ, Bʹ and Cʹ. (c) Reflect A, B, C, Aʹ, Bʹ and Cʹ about x-axis and name them as Aʺ, Bʺ, Cʺ, Aʺʹ, Bʺʹ and Cʺʹ respectively. (d) Join A, B, C, Cʺ, Bʺ, Aʺ, Aʺʹ, Bʺʹ, Cʺʹ, Cʹ, Bʹ, Aʹ and A to form a closed figure.

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(a) Plot A(1, 2), B(1, 1), C(2, 1). (b) Reflection in the y-axis changes (x, y) to (−x, y): Aʹ(−1, 2), Bʹ(−1, 1), Cʹ(−2, 1). (c) Reflection in the x-axis changes (x, y) to (x, −y): Aʺ(1, −2), Bʺ(1, −1), Cʺ(2, −1), Aʺʹ(−1, −2), Bʺʹ(−1, −1), Cʺʹ(−2, −1). (d) Joining A(1, 2) → B(1, 1) → C(2, 1) → Cʺ(2, −1) → Bʺ(1, −1) → Aʺ(1, −2) → Aʺʹ(−1, −2) → Bʺʹ(−1, −1) → Cʺʹ(−2, −1) → Cʹ(−2, 1) → Bʹ(−1, 1) → Aʹ(−1, 2) → A gives a closed figure in the shape of a plus sign (cross), symmetric about both axes.

ICSE 2023 sample paper · 1 mark

The point A(p, q) is invariant about x = p under reflection. The coordinates of it's image A' is:

  1. A'(p, -q)
  2. A'(-p, q)
  3. A'(p, q)
  4. A'(-p, -q)
Show answer
A'(p, q)

ICSE 2023 board paper · 5 marks

Use graph sheet to answer this question. Take 2 cm = 1 unit along both the axes. (a) Plot A, B, C where A(0, 4), B(1, 1) and C(4, 0) (b) Reflect A and B on the x-axis and name them as E and D respectively. (c) Reflect B through the origin and name it F. Write down the coordinates of F. (d) Reflect B and C on the y-axis and name them as H and G respectively. (e) Join points A, B, C, D, E, F, G, H and A in order and name the closed figure formed.

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(a) A(0, 4), B(1, 1), C(4, 0) plotted. (b) Reflection in the x-axis: (x, y) → (x, -y). E = (0, –4), D = (1, –1). (c) Reflection through the origin: (x, y) → (-x, -y). F = (–1, –1). (d) Reflection in the y-axis: (x, y) → (-x, y). H = (–1, 1), G = (–4, 0). (e) Joining A(0, 4), B(1, 1), C(4, 0), D(1, –1), E(0, –4), F(–1, –1), G(–4, 0), H(–1, 1) and back to A gives a four-pointed star (a concave octagon).

ICSE 2023 sample paper · 5 marks

Use graph sheet for this question. Take 2 cm = 1 unit along the axes. Plot the Δ OAB, where O(0, 0), A(3, -2), B(2, -3). (a) Reflect the Δ OAB through the origin and name it as Δ OA'B'. (b) Reflect the Δ OA'B' on the y-axis and name it as Δ OA''B''. (c) Reflect the Δ OA'B' on the x-axis and name it as Δ OA'''B'''. (d) Join the points AA''B''B'A'A'''B'''B and give the geometrical name of the closed figure so formed.

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Plot O(0, 0), A(3, −2), B(2, −3) and join them. (a) Reflection in the origin: (x, y) → (-x, -y): A'(−3, 2), B'(−2, 3). (b) Reflection of ΔOA'B' in the y-axis: (x, y) → (-x, y): A''(3, 2), B''(2, 3). (c) Reflection of ΔOA'B' in the x-axis: (x, y) → (x, -y): A'''(−3, −2), B'''(−2, −3). (d) Joining A(3, −2), A''(3, 2), B''(2, 3), B'(−2, 3), A'(−3, 2), A'''(−3, −2), B'''(−2, −3), B(2, −3) and back to A gives a closed figure with 8 sides: an octagon.

ICSE 2022 sample paper · 1 mark

The point (3,0) is invariant under reflection in:

  1. The origin
  2. x-axis
  3. y-axis
  4. both x and y axes
Show answer
x-axis