Grade by Grade
E. Spatial Sense · Spatial Sense: Angles, Tessellations, Transformations, Scale, Pythagoras and Composite Measurement

Reflections, Rotations and Combined Transformations

E1.4 A1About 50 minutes
Learning goals
  • I can reflect shapes in the lines y = x and y = −x and write the coordinate rule.
  • I can rotate a point or shape about a centre that is not the origin.
  • I can perform a sequence of transformations, explain whether the order matters, and predict where the image will land.
  • I can identify a single transformation that has the same result as two transformations.

Words to know

image
The result of a transformation. Image points are named with primes: A becomes A′ ("A prime").
coordinate rule
A rule that says what happens to the coordinates of every point, like (x, y) → (x + 3, y − 2).
line y = x
The diagonal line through the origin made of all points whose two coordinates are equal, like (2, 2) and (−3, −3).
centre of rotation
The fixed point that a shape turns around. It does not move.
congruent
Same size and same shape. Translations, reflections and rotations always give a congruent image.
sequence of transformations
Two or more transformations done one after the other. The image of the first is the starting shape for the second.

Remember from Grade 7

In Grade 7 you wrote coordinate rules for transformations on the Cartesian plane:

  • translation a right and b up: (x, y) → (x + a, y + b)
  • reflection in the x-axis: (x, y) → (x, −y); in the y-axis: (x, y) → (−x, y)
  • rotation about the origin: 90° clockwise (x, y) → (y, −x); 90° counterclockwise (x, y) → (−y, x); 180° (x, y) → (−x, −y)

You also reflected in vertical and horizontal lines like x = 2. In Grade 8 you'll reflect in diagonal lines, rotate about any point, and string transformations together.

Let's start with an animator

An animator wants a character to cartwheel across the screen: turn a quarter turn around its hand, then around its foot, then slide along. Each move is a transformation with its own centre. To program it, the animator has to know exactly where every point lands after each step, and whether doing the steps in a different order changes the result.

xy−1123456−1123456y = xABCA′B′C′xy−2−112345−112345CPP′Reflection in y = x90° turn about C(1, 1)
Left: triangle ABC reflected in the dashed line y = x. A(1, 3) → A′(3, 1), B(3, 4) → B′(4, 3), C(1, 5) → C′(5, 1). Right: P(4, 3) rotated 90° counterclockwise about C(1, 1) lands on P′(−1, 4). The two dashed segments from C are the same length and meet at a right angle.

Reflecting in y = x and y = −x

The line y = x goes diagonally through the origin at 45°. Reflecting in it swaps the coordinates:

Reflection in y = x: (x, y) → (y, x)

The line y = −x goes diagonally the other way, through (−1, 1) and (1, −1). Reflecting in it swaps the coordinates and changes both signs:

Reflection in y = −x: (x, y) → (−y, −x)

Rotating about a centre that is not the origin

The Grade 7 rotation rules only work around the origin. For any other centre C, use move, turn, move back:

  1. Move: subtract C's coordinates from the point, as if C were the origin.
  2. Turn: use the origin rule (90° clockwise, 90° counterclockwise or 180°).
  3. Move back: add C's coordinates again.
Worked example

Example 1: Reflect in y = x

Reflect triangle A(1, 3), B(3, 4), C(1, 5) in the line y = x.

  1. Swap the coordinates of each vertex: (x, y) → (y, x).
  2. A(1, 3) → A′(3, 1). B(3, 4) → B′(4, 3). C(1, 5) → C′(5, 1).
  3. Check on the grid: each point and its image are the same distance from the line y = x, on opposite sides, and the segment joining them crosses the line at a right angle.

A′(3, 1), B′(4, 3), C′(5, 1) (see the left grid). The triangle was above the line y = x, so its image is below it.

Worked example

Example 2: Rotate about a point

Rotate P(4, 3) 90° counterclockwise about C(1, 1).

  1. Move: subtract C. (4 − 1, 3 − 1) = (3, 2).
  2. Turn: 90° counterclockwise about the origin is (x, y) → (−y, x). (3, 2) → (−2, 3).
  3. Move back: add C. (−2 + 1, 3 + 1) = (−1, 4).
  4. Check: from C to P is 3 right and 2 up. From C to P′ is 2 left and 3 up. Both are the same length, and they are at right angles (see the right grid).

P′(−1, 4)

Worked example

Example 3: Does the order matter?

Start with (2, 1). Plan A: translate 3 units right, then reflect in the y-axis. Plan B: reflect in the y-axis, then translate 3 units right. Do they end in the same place?

  1. Plan A: (2, 1) → (5, 1) → (−5, 1).
  2. Plan B: (2, 1) → (−2, 1) → (1, 1).
  3. The images are different.

No. Plan A ends at (−5, 1) and Plan B at (1, 1). For a sequence of transformations, the order can matter, so always do them in the order given.

Worked example

Example 4: Two reflections make one translation

Reflect (0, 2) in the line x = 1, then reflect the image in the line x = 4. What single transformation does the same thing?

  1. (0, 2) is 1 unit left of x = 1, so its image is 1 unit right of it: (2, 2).
  2. (2, 2) is 2 units left of x = 4, so its image is 2 units right of it: (6, 2).
  3. The point moved 6 units right. Try another point, (3, −1): → (−1, −1) → (9, −1). Again 6 units right.
  4. The lines are 3 units apart, and the point moved 2 × 3 = 6 units.

A translation 6 units right: (x, y) → (x + 6, y). Reflecting in two parallel lines is the same as translating by twice the distance between them.

−112345670123x = 1x = 4A(0, 2)A′(2, 2)A″(6, 2)1122moved 6 right = 2 × 3 (the lines are 3 apart)
Example 4 on a grid. Reflecting A in x = 1 and then in x = 4 moves it 6 units right. The two mirror lines are 3 units apart, and the point moves twice that distance.

Predict, then check

Before you calculate, predict: Which quadrant will the image be in? Will it face the same way? Then check:

  • Translations, reflections and rotations keep lengths and angles, so the image is congruent. Measure one side to check.
  • A reflection reverses the order of the vertices (clockwise becomes counterclockwise). Translations and rotations don't.
  • For a rotation, the centre and each point stay the same distance apart.
Hands-on activity

Tracing-paper transformations

You need: graph paper (or draw a grid with a ruler), tracing paper or parchment paper, a pencil and a pushpin or the tip of a sharp pencil (careful with the point).

  1. Draw x- and y-axes from −8 to 8. Plot triangle A(1, 3), B(3, 4), C(1, 5).
  2. Reflection in y = x: draw the line y = x. Trace the triangle and the line. Flip the tracing paper over so the traced line lands back on y = x. Mark where the vertices land. Compare with the rule (x, y) → (y, x).
  3. Rotation about C(1, 1): trace the triangle and the point C(1, 1). Hold the tracing paper down at C with the pencil tip and turn it a quarter turn counterclockwise (an axis drawn on the tracing paper should line up with the other axis). Mark the new vertices. Check them with move-turn-move back.
  4. Order test: with a partner, each pick a different order for "translate 3 right" and "reflect in the x-axis". Compare images.

What to look for: the tracing results match the rules. Translation and reflection in the x-axis happen to give the same result in either order here (the slide is along the mirror line), but translation and reflection in the y-axis would not (Example 3).

Math talk: "When does the order matter, and when doesn't it?"

Just for fun

Rotate a point 90° about the same centre four times and it ends up right back where it started. Mathematicians call that a full turn of 360°. Dogs call it getting ready to lie down.

Your turn

1

Reflect (3, −2) in the line y = x.

2

Reflect (−4, 1) in the line y = −x.

3

Rotate (2, 5) 180° about the origin.

4

Rotate (3, 1) 90° clockwise about the origin.

5

Rotate (5, 2) 180° about (1, 1).

6

Translate (−2, 4) using (x, y) → (x + 5, y − 6), then reflect the image in the x-axis.

7

(4, 1) moves to (1, 4). Name a single reflection that does this.

More practice: warm-up

1

Write the rule for a translation 4 units left and 2 units up.

2

Reflect (5, 3) in the x-axis.

3

Reflect (5, 3) in the y-axis.

4

Is the image of a rotation congruent to the original shape?

5

Reflect (6, −1) in the line y = x.

More practice: on your own

1

Rotate the triangle (1, 1), (3, 1), (1, 4) 90° counterclockwise about the origin. Which quadrant is the image in?

2

Reflect (2, 5) in the horizontal line y = 1.

3

Rotate Q(4, 1) 90° clockwise about (2, −1).

4

Reflect (3, 2) in the x-axis, then reflect the image in the y-axis. What single transformation gives the same result?

5

A(1, 2) → A′(−2, 1) and B(4, 2) → B′(−2, 4). Identify the transformation.

Challenge

1

Reflect a shape in the line x = 2 and then in the line x = 5. What single transformation does the same thing? Check with the point (0, 0).

2

A rotation of 180° moves (1, 3) to (5, −1). Where is the centre of rotation?

3

Test with (2, 1): is "rotate 90° clockwise about the origin, then reflect in the x-axis" the same as doing them in the opposite order? What single transformation is the first order?

Check your understanding

Quiz

1.

Reflect (2, 7) in the line y = x.

2.

Which transformation has the rule (x, y) → (−x, −y)?

3.

Rotate (0, 3) 90° clockwise about the origin.

4.

A shape is reflected in the line x = 1 and then in the line x = 3. What single transformation does the same thing?

5.

The images of a shape under translations, reflections and rotations are always:

6.

A rotation of 180° about (2, 2) moves (3, 2) to:

For parents and teachers

Big idea: The Ontario curriculum (E1.4) asks Grade 8 students to describe and perform translations, reflections, rotations and dilations on a Cartesian plane, and predict the results. Grade 7 introduced coordinate rules; Grade 8 adds diagonal mirror lines, rotations about any point, and sequences of transformations. Dilations are in the next lesson.

Common difficulties:

  • Using an origin rule for a different centre. Remind your child of move, turn, move back.
  • Mixing up clockwise and counterclockwise rules. A quick check: rotating (0, 1) (straight up) 90° clockwise should give (1, 0) (straight right).
  • Doing a sequence in the wrong order. The order can change the answer.

How to help: Tracing paper (or parchment paper) makes transformations physical and lets your child check answers on their own. Ask them to predict the quadrant before they calculate. This lesson also builds A1 skills: making conjectures, testing them with examples and generalizing.