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Year 6 Maths 2D Shapes, Position & Transformations

Translate, reflect & rotate shapes

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Theory

A translation slides a shape, a reflection flips it over a mirror line, and a rotation turns it about a centre. All three keep the shape and size the same, so the image is congruent to the original. This Year 6 maths page shows how to carry out and describe each transformation on the number plane, find where points move to, and combine two transformations.

A translation slides every point the same distance in the same direction. Describe it as "\(4\) units left and \(5\) units down".

A reflection flips a shape over a mirror line. Each image point is the same distance from the line, on the other side. Reflecting in the \(x\)-axis changes the sign of \(y\); reflecting in the \(y\)-axis changes the sign of \(x\).

A rotation turns a shape about a centre of rotation by an angle: a quarter turn is \(90^\circ\), a half turn \(180^\circ\), clockwise or anticlockwise.

The image is the shape after the move. It is congruent (same shape and size) to the original. Only a reflection makes it face the opposite way.

Translation of a rectangle A number plane from negative 6 to 6. A shaded rectangle with corners (2, 3), (5, 3), (5, 5), (2, 5) and its dashed image with corners (negative 2, negative 2), (1, negative 2), (1, 0), (negative 2, 0): the rectangle has slid 4 units left and 5 units down. −6−4−2246 −6−4−2246 xy
Translation: every corner moves 4 units left and 5 units down.
Reflection in the x-axis A number plane with the x-axis drawn in red as the mirror line. A shaded triangle with corners (1, 2), (5, 2), (1, 5) and its dashed image with corners (1, negative 2), (5, negative 2), (1, negative 5). −6−4−2246 −6−4−2246 xy
Reflection in the \(x\)-axis: \((x, y)\) becomes \((x, -y)\).
Quarter turn anticlockwise about the origin A number plane with a red dot at the origin. A shaded triangle with corners (1, 1), (4, 1), (4, 3) and its dashed image after a quarter turn anticlockwise about the origin, with corners (negative 1, 1), (negative 1, 4), (negative 3, 4). −6−4−2246 −6−4−2246 xy
A quarter turn (\(90^\circ\)) anticlockwise about the origin.

Transformations on the number plane:

MoveExample
Translation\(4\) left, \(5\) down
Reflect in \(x\)-axis\((5, 2) \to (5, -2)\)
Reflect in \(y\)-axis\((2, 4) \to (-2, 4)\)
Half turn about \((0, 0)\)\((-4, 1) \to (4, -1)\)
Same shape, same size. Transformations never change side lengths or angles.
Facing the other way? Only a reflection turns a shape into its mirror image.

Describing a transformation

  1. Same way round, just moved? Translation: give the distance across and up/down.
  2. Flipped? Reflection: name the mirror line.
  3. Turned? Rotation: give the centre, the angle and the direction.

Finding an image point

  1. Translation: add or subtract the moves from \(x\) and \(y\).
  2. Reflection: count the distance to the mirror line and go the same distance past it.
  3. Rotation: turn the shape (tracing paper helps) and read the new corners.
Example 1 — Describe a translation
Use the first diagram. Describe the translation fully.
Solution

The corner \((5, 5)\) moves to \((1, 0)\).

\(5 - 4\)\(=\)\(1\)
\(5 - 5\)\(=\)\(0\)
4 left ,5 down

4 units left and 5 units down.

Example 2 — Reflect in the x-axis
Use the second diagram. Where does the corner \((5, 2)\) go?
Solution

It is \(2\) above the mirror line, so the image is \(2\) below: \((5, -2)\).

Example 3 — Quarter turn
Use the third diagram.
(i) Through how many degrees does the triangle turn?
(ii) Where does the corner \((4, 3)\) go?
Solution

(i) \(90^\circ\) anticlockwise
(ii) \((-3, 4)\)

Example 4 — Half turn
The point \((-4, 1)\) is rotated a half turn about the origin. Where is its image?
Solution

A half turn sends it to the opposite side of the origin: \((4, -1)\).

Example 5 — Name it
A shape and its image are the same way round but in a new place. Which transformation is it?
Solution

A translation.

Example 6 — Two moves
The point \((2, 4)\) is reflected in the \(y\)-axis and then translated \(3\) units down. Where does it end up?
Solution
\(4 - 3\)\(=\)\(1\)
(2,4)→(−2,4)→(−2,1)

\((2, 4) \to (-2, 4) \to (-2, 1)\).

Example 7 — Congruent
A rectangle \(9\) cm by \(4\) cm is rotated a quarter turn, then reflected.
(i) How long are the sides of the final image?
(ii) What is its area?
Solution
\(9 \times 4\)\(=\)\(36\)
9×4=36

(i) \(9\) cm and \(4\) cm
(ii) \(36\,\text{cm}^2\), the same as before

Example 8 — Find the mistake
Zac says the point \((3, 2)\) reflected in the \(y\)-axis goes to \((3, -2)\). Explain the mistake.
Solution

Reflecting in the \(y\)-axis changes the sign of the \(x\)-coordinate, not the \(y\)-coordinate, so the image is \((-3, 2)\). The point \((3, -2)\) is the reflection in the \(x\)-axis.

Common pitfalls

Calling a flip a turn. If the image faces the opposite way, it is a reflection.
Half-describing. A translation needs both directions; a rotation needs centre, angle and direction.
Changing the size. The image is always the same size.
Reflecting the wrong coordinate. In the \(x\)-axis, \(y\) changes sign; in the \(y\)-axis, \(x\) changes sign.

Frequently asked questions

What is a translation?

A slide: every point moves the same distance in the same direction.

What happens to (5, 2) when it is reflected in the x-axis?

It becomes \((5, -2)\).

How many degrees is a quarter turn?

\(90^\circ\).

Does a transformation change the size of a shape?

No. Translations, reflections and rotations keep the shape and size.

How can I tell a reflection from a rotation?

A reflection makes a mirror image that faces the other way; a rotation does not.

More in 2D Shapes, Position & Transformations
Plot & locate points on the coordinate plane (four quadrants)
20 questions