Year 10 Standard · Unit 5 Solids, similarity and transformations · Lesson 5 of 6

What must you say so that someone else can draw the same image?

Each transformation has its own list of details. Say too little and two people draw different images; say it in full and the image is fixed. Translations, rotations and reflections keep a shape congruent; an enlargement does not.

Geometry and TrigonometryNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: a point (3, 5) is reflected in the x-axis. What are the coordinates of its image?
    Answer(3, −5). A reflection in the x-axis keeps x and changes the sign of y.
  2. From Year 9: a shape is enlarged by scale factor 2 from the origin. A point on it is (4, 3). Where is its image?
    Answer(8, 6), because both coordinates are multiplied by 2.
  3. From lesson 4: two similar solids have lengths in the ratio 1 : 3. What is the ratio of their surface areas?
    Answer1 : 9, because areas go with the square of the length ratio.
Words used in this lesson
translation
a slide: every point moves the same distance in the same direction. For example: top 3, bottom −2.
rotation
a turn about a fixed point through an angle. For example: a wheel turning on its axle.
reflection
a flip in a mirror line. For example: in the line y = x.
enlargement
a change of size from a centre by a scale factor. For example: scale factor 2 from the origin.
isometry
a transformation that keeps every distance. For example: a rotation.
column vector
two numbers written one above the other for across and up. For example: top 0, bottom 5: five squares straight up.

A tile that landed in the wrong place

A mosaic maker in Fez has a triangular tile in her pattern book. A friend copies the pattern from the instruction “turn the tile 90° clockwise”, but the friend's tile lands in a different position from hers.

Predict first: what is missing from the instruction: the angle, the direction, the centre, or something else? Write down a reason, then use the first figure and press Rotate.

3
−2
(4, −1)image of A (1, 1)
(4, 0)image of C (1, 2)

Translation by the column vector with top 3 and bottom −2: A (1, 1) goes to (4, −1), B (3, 1) goes to (6, −1) and C (1, 2) goes to (4, 0).

Show me the steps
  1. Translate with p = 3 and q = −2: every point moves 3 right and 2 down. A (1, 1) goes to (4, −1), B to (6, −1), C to (4, 0).
  2. Press Rotate with p = 0 and q = 0: the triangle turns 90° clockwise about the origin. A goes to (1, −1), B to (1, −3), C to (2, −1).
  3. Keep Rotate and move the centre to (2, 1): the same angle and direction now put A at (2, 2), B at (2, 0) and C at (3, 2). The centre changed, so the image changed.
  4. A rotation needs three things: the angle, the direction, and the centre. A translation needs a column vector and nothing else.

Four transformations, four descriptions. A translation needs a column vector (across on top, up on the bottom). A rotation needs the angle, the direction and the centre. A reflection needs the mirror line. An enlargement needs a scale factor and a centre.

Translations, rotations and reflections are isometries: they keep every distance, so the image is congruent to the object. An enlargement keeps the shape but changes the size, so the image is similar, not congruent.

−2
3
(−5, 1)image of A (1, 1)
(−5, 2)image of C (1, 2)

Reflection in the line x = −2: A (1, 1) goes to (−5, 1), B (3, 1) goes to (−7, 1) and C (1, 2) goes to (−5, 2).

Show me the steps
  1. Reflection in x = −2: A (1, 1) goes to (−5, 1), B to (−7, 1), C to (−5, 2). Each point lands as far on the other side of the mirror as it started.
  2. Press y = q with q = 3: A goes to (1, 5), B to (3, 5) and C to (1, 4).
  3. Press y = x: A stays at (1, 1), B goes to (1, 3) and C goes to (2, 1). The coordinates swap.
  4. Press x = p with p = 1: A and C lie on the mirror, so they do not move. A point on the mirror line is its own image.

Say it in full. A translation is a column vector, not a pair of coordinates. A rotation without its centre is not a description. A reflection in y = x swaps the coordinates, but a reflection in x = 3 does not: it works from the mirror.

Worked examples

1. A translation maps A (−3, 2) to A′ (4, −1). Describe it.

  1. Across: 4 − (−3) = 7. Up: −1 − 2 = −3.
  2. It is a translation by the column vector with top 7 and bottom −3.

2. Rotate the point (3, −1) by 90° clockwise about the origin.

  1. A 90° clockwise turn about the origin sends (x, y) to (y, −x).
  2. (3, −1) goes to (−1, −3). Check: it is still the same distance from the origin, √10.

3. Enlarge the point (4, 1) by scale factor 2 from the centre (2, 0).

  1. The point is 2 right and 1 up from the centre. Double that: 4 right and 2 up.
  2. The image is (2 + 4, 0 + 2) = (6, 2).

4. A segment from P (3, 4) to Q (3, 1) is mapped to P′ (−1, 0) and Q′ (−1, 3). Describe the transformation.

  1. The segment stayed vertical and its direction reversed, so the shape was turned half a turn. The centre is the midpoint of P and P′: (1, 2).
  2. Check with Q and Q′: the midpoint is (1, 2) as well. It is a rotation of 180° about (1, 2).

Your go. The first step is given. Reflect the point (6, 3) in the line y = 1.

  1. The point is 2 above the mirror line, since 3 − 1 = 2.
  2. The image must be 2 below the line. Work out its y-coordinate; the x-coordinate does not change.
Check the answer

The image is 2 below y = 1, so y = −1: the image is (6, −1). By the formula, 2 × 1 − 3 = −1.

Check before you go on

1. A column vector is written with the movement across on top and the movement up at the bottom. A triangle moves 4 units left and 1 unit up. Which column vector describes this translation?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. The point P (6, −2) is translated by the column vector with top −4 and bottom 5. What is the x-coordinate of the image?

3. The point (2, 5) is reflected in the line x = 6. What is the x-coordinate of the image?

Level 2 · Building

4. The point (2, 5) is rotated 90° clockwise about the origin. What is the y-coordinate of the image?

5. The point P (4, 1) is rotated 90° clockwise about the centre (2, 3). What is the x-coordinate of the image?

6. The point (3, 2) is enlarged by scale factor 3 from the centre (1, 1). What is the x-coordinate of the image?

Level 3 · Stretch

7. A translation maps A (2, 6) to A′ (−1, 4). What is the top number of the column vector?

8. The point (1, 5) is reflected in the line y = x + 2. What is the x-coordinate of the image?

Something to think about

In the first figure, a rotation of 90° clockwise about the origin sends (1, 1) to (1, −1). Is there a translation that does the same thing to the whole triangle? Try it.

One way to think about it

No. The rotation sends B (3, 1) to (1, −3), so B moves 2 left and 4 down, while A moves 0 right and 2 down. A translation moves every point by the same vector, but a rotation moves different points by different amounts. The one exception is a rotation of 360°, which is the same as no move at all.

Exit ticket

9. A segment from A (5, 2) to B (5, 4) is mapped to A′ (2, −5) and B′ (4, −5). Which is a full description of the transformation?