Practice Maths

L62 — Reflections

Key Terms

reflection
A transformation that flips a shape over a mirror line. Each point and its image are the same perpendicular distance from the mirror line, on opposite sides.
mirror line
The line of reflection. Points on the mirror line do not move. Common mirror lines: x-axis, y-axis, y = x.
image
The result after the transformation. Image points are labelled with a prime: A becomes A′.
orientation
The direction a shape faces. Reflections reverse orientation — the image is a mirror image (flipped).
y = x x-axis y-axis 1 2 3 −1 −2 −3 1 2 3 −1 −2 −3 A(3, 2) A′(3, −2) x-axis reflection A′(−3, 2) y-axis reflection A′(2, 3) y = x reflection

Blue = original A(3, 2). Each colour shows one type of reflection.

Reflection Rules

  • Over the x-axis (horizontal line): (x, y) → (x, −y) — x stays, y flips sign
  • Over the y-axis (vertical line): (x, y) → (−x, y) — y stays, x flips sign
  • Over y = x (diagonal line): (x, y) → (y, x) — swap x and y
Hot Tip: x-axis: x stays, y flips — (x, y) → (x, −y)  •  y-axis: y stays, x flips — (x, y) → (−x, y)  •  y = x: swap x and y — (x, y) → (y, x)

Worked Example 1 — Reflect over x-axis

Reflect (3, 4) over the x-axis.

Rule: (x, y) → (x, −y). So (3, 4) → (3, −4).

Worked Example 2 — Reflect over y-axis

Reflect (3, 4) over the y-axis.

Rule: (x, y) → (−x, y). So (3, 4) → (−3, 4).

Worked Example 3 — Reflect over y = x

Reflect (3, 4) over the line y = x.

Rule: (x, y) → (y, x). So (3, 4) → (4, 3).

Worked Example 4 — Reflect a Shape

Reflect triangle (1, 2), (3, 2), (2, 5) over the y-axis.

(1, 2) → (−1, 2)

(3, 2) → (−3, 2)

(2, 5) → (−2, 5)

x y (3,4) (3,−4) x-axis (−3,4) y-axis

Blue = original. Red = reflected over x-axis. Green = reflected over y-axis.

What Is a Reflection?

A reflection flips a shape across a mirror line (also called the line of reflection or axis of reflection). The image is a mirror image of the original — same shape and size, but flipped. Think of a reflection in a still lake: the tree on the bank appears upside down in the water, at the same distance from the water's surface.

The mirror line can be horizontal, vertical, or diagonal. Each point in the original shape maps to a corresponding point in the image, with the mirror line exactly in the middle between them.

The Key Property: Equidistant Points

The most important property of reflections: each point and its image are the same distance from the mirror line, and the line connecting them is perpendicular (at 90°) to the mirror line.

  • If point A is 3 units to the left of a vertical mirror line, then A′ is 3 units to the right of the same line.
  • If a point is ON the mirror line, it maps to itself (it doesn't move).

Reflecting in Horizontal and Vertical Lines

  • Reflecting in the x-axis: The y-coordinate changes sign. Point (3, 4) reflects to (3, −4).
  • Reflecting in the y-axis: The x-coordinate changes sign. Point (3, 4) reflects to (−3, 4).
  • Reflecting in the line y = 3: Keep the x-coordinate; reflect the y-coordinate about 3. Point (2, 1) → distance from y=3 is 2 → image is (2, 5).

Reflecting in the Line y = x

Reflecting in the line y = x (the diagonal line at 45° through the origin) swaps the x and y coordinates:

  • Point (2, 5) reflects to (5, 2)
  • Point (−3, 1) reflects to (1, −3)
Key tip: To reflect a shape on a grid, reflect each vertex (corner) one at a time, then connect the image points in the same order. Count the number of squares from each point to the mirror line, then count the same number of squares on the other side. This counting method is the most reliable way to avoid errors.

  1. Reflect over the x-axis Fluency

    Reflect each point over the x-axis (the orange line). Rule: (x, y) → (x, −y). Each point and its image are the same distance from the x-axis, on opposite sides.

    x-axis 2−2 5−5 A(2, 5) A′(2, −5)
    (a) Reflect (2, 5)
    x-axis 4 −33 B(4, −3) B′(4, 3)
    (b) Reflect (4, −3)
    x-axis −1 7−7 C(−1, 7) C′(−1, −7)
    (c) Reflect (−1, 7)
    x-axis 6−6 D(0, 6) D′(0, −6)
    (d) Reflect (0, 6)

  2. Reflect over the y-axis Fluency

    Reflect each point over the y-axis (the orange line). Rule: (x, y) → (−x, y). The y-coordinate stays the same; the x-coordinate changes sign.

    y-axis 3−3 4 A(3, 4) A′(−3, 4)
    (a) Reflect (3, 4)
    y-axis −55 2 B(−5,2) B′(5, 2)
    (b) Reflect (−5, 2)
    y-axis 7−7 −1 C(7, −1) C′(−7,−1)
    (c) Reflect (7, −1)
    y-axis 3 D(0, 3) on mirror line — stays put!
    (d) Reflect (0, 3) — note: x = 0 is on the y-axis

  3. Reflect over y = x Fluency

    Reflect each point over the line y = x (the orange diagonal). Rule: (x, y) → (y, x). The coordinates swap.

    y = x 27 72 A(2, 7) A′(7, 2)
    (a) Reflect (2, 7)
    y = x 41 41 B(4, 1) B′(1, 4)
    (b) Reflect (4, 1)
    y = x −35 5−3 C(−3, 5) C′(5, −3)
    (c) Reflect (−3, 5)
    y = x 6 6 D(0, 6) D′(6, 0)
    (d) Reflect (0, 6)

  4. Reflect Shapes Fluency

    Find the image vertices of each shape after the given reflection. Blue = original; red dashed = image.

    x-axis 1 4 5 2 −2 −5 A B C A′ B′ C′
    (a) ABC over x-axis
    y-axis −1 2 3 6 P Q P′ Q′ R = R′
    (b) PQR over y-axis
    x-axis 1 4 3 1 −1 −3 D E F G D′ E′ F′ G′
    (c) DEFG over x-axis
    y = x J(2,3) K(5,1) L(3,6) J′(3,2) K′(1,5) L′(6,3)
    (d) JKL over y = x

  5. Identify the Mirror Line Understanding

    Each diagram shows a point (blue) and its image (red). Identify the mirror line and write the reflection rule used.

    3 5 −5 A(3, 5) A′(3, −5)
    (a) What is the mirror line?
    −2 2 4 B(−2, 4) B′(2, 4)
    (b) What is the mirror line?
    7 3 3 7 C(7, 3) C′(3, 7)
    (c) What is the mirror line?
    4 −4 −2 D(4, −2) D′(−4, −2)
    (d) What is the mirror line?

  6. Reflections in Context Problem Solving

    y-axis −2 −5 4 7 A′ B′ C′ ?
    (a) A′B′C′ given — find original ABC
    centre 6 3 (6, 3) (?)
    (b) Find the mirror image of (6, 3)
    T1 T2 T3 1 3 2
    (c) Kaleidoscope pattern
    1. Triangle A′B′C′ has vertices A′(−2, 4), B′(−5, 4), C′(−3, 7). It was created by reflecting triangle ABC over the y-axis. Find A, B, and C.
    2. The flag of a country has a design that is symmetric about a vertical centre line. If a point on the design is at (6, 3) and the centre line is the y-axis, where is its mirror image?
    3. A kaleidoscope image is created by reflecting a triangle with vertices (1, 0), (3, 0), (2, 2) over the x-axis (T2) and over the y-axis (T3). List all three sets of vertices and describe the pattern.
    4. Point P(a, b) is reflected over the x-axis to get P′, then P′ is reflected over the y-axis to get P′′. Write the coordinates of P′′ in terms of a and b. What single transformation gives the same result?

  7. Mixed Reflection Practice Fluency

    For each point, apply all three reflections and record the image coordinates: (i) over the x-axis, (ii) over the y-axis, (iii) over y = x. Blue = original • Red = x-axis image • Green = y-axis image • Purple = y = x image

    y=x 3 1 1 3 (3,1) (i)? (ii)? (iii)?
    (a) Original (3, 1)
    y=x −4 4 2 −4 (−4,2) (i)? (ii)? (iii)?
    (b) Original (−4, 2)
    y=x 5 −5 5 −5 (5,−5) (i)? (ii)? (iii)?
    (c) Original (5, −5)
    y=x 4 4 −4 (i)? (iii)? (0,4) ii: stays!
    (d) Original (0, 4)

  8. Reflection on a Grid — Describe and Apply Understanding

    4 −4 3 A(4, 3) A′(−4, 3)
    (a) Name the mirror line
    2 5 −5 B(2, 5) B′(2, −5)
    (b) Name the mirror line
    T1 T2 T3
    (c) T1 → x-axis → T2 → y-axis → T3
    1. Point A(4, 3) is reflected to A′(−4, 3). What is the mirror line? Write the rule used.
    2. Point B(2, 5) is reflected to B′(2, −5). What is the mirror line? Write the rule used.
    3. Triangle with vertices (1, 2), (3, 4), (5, 1) is reflected over the x-axis (T2). Then the image is reflected over the y-axis (T3). List the final vertices. What single transformation gives the same overall result?
    4. Explain in your own words why a reflection changes orientation (the shape appears flipped) but not size.

  9. Reflections — Find the Missing Coordinate Understanding

    Each diagram shows a point (blue) and its image (red), but one coordinate is missing. Find the ? and state the mirror line.

    x-axis 3 5 (3, 5) (3, ?)
    (a) Find the missing y-coordinate
    y-axis −4 4 7 (−4, ?) (4, 7)
    (b) Find the missing y-coordinate
    y = x 6 6 3 (?, 6) (6, 3)
    (c) Find the missing x-coordinate
    y = x 2 −3 −3 2 (2, −3) (?, 2)
    (d) Find the missing x-coordinate

  10. Reflections in the Real World Problem Solving

    y-axis y = 0 3 5 −3 −5 8 5 given find?
    (a) Symmetric building
    4 3 orig y-axis refl x-axis refl
    (b) Triangle reflections
    lake (x-axis) 5 6 −6 tree (5,6) image (5,?) path?
    (c) Tree and lake reflection
    centre 3 −3 2 (3,2) (−3,2) (2,2) +5
    (d) Dancer reflection + walk
    1. An architect draws half of a symmetric building. The right half has corners at (0, 0), (0, 8), (3, 8), (3, 5), (5, 5), (5, 0). The mirror line is the y-axis. Find the left-half corner coordinates.
    2. A pattern is made by reflecting a right triangle with legs along the axes. The triangle has vertices at (0, 0), (4, 0), (0, 3). Reflect it over the y-axis. Then reflect the original over the x-axis. List all vertices of both reflections. What shape do the original and the y-axis reflection together form?
    3. A lake is represented by the x-axis. A tree is located at position (5, 6) on the map. Find the mirror image coordinates (reflected over the x-axis). If a path is built straight from the tree to its reflection, how long is the path?
    4. A dancer stands at (3, 2) and reflects over the y-axis (the centre line). Where do they stand after the reflection? If they then walk right 5 units, what are the final coordinates?
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