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KS3 Maths · Year 8Chapter 11 · Transformations and similarity
Year 8 · Chapter 11

Transformations and similarity

By the end of this chapter you will be able to

  • Translate a shape using a column vector, and find the vector that moves one shape to another
  • Reflect a shape in a mirror line x = a, y = b, y = x or y = −x
  • Rotate a shape by 90°, 180° or 270° about the origin, and about a given point, clockwise or anticlockwise
  • Enlarge a shape from a centre by a scale factor, and find the scale factor from two lengths
  • Recognise congruent and similar shapes, and use a scale factor to find a missing length

You should already know

  • Coordinates in all four quadrants, and plotting points (Year 7 and Year 8, Chapter 7)
  • Angles and the names of quadrilaterals and triangles (Year 7, Chapter 9)
  • Multiplying and dividing decimals and fractions (Year 7 and Year 8, Chapter 3)

Where this leads

  • Enlargements with negative and fractional scale factors, and the effect on area (Year 9)
  • Congruence and similarity proofs in triangles (GCSE)
  • Vectors and combining transformations (GCSE)

National Curriculum for England, Key Stage 3 mathematics, Year 8. Content: Describe and carry out translations (with vectors), reflections in x = a, y = b, y = x and y = −x, rotations about a point, and enlargements with a centre and scale factor, Identify congruent shapes, Use scale factors to find lengths in similar shapes.

Exercise P  Do you remember?

Non-calculator
1
FPlot the point (3, 2). How far along the x-axis is it?
2
FWhich way is a clockwise turn: to the left or to the right?
3
FWhat is 2 × 5 − 3?
4
FA triangle has sides 3 cm, 4 cm and 5 cm. Is it a right-angled triangle? Just say yes or no.
5
FWork out 4 ÷ 2 ÷ 2.
11.1

Translations with vectors

Year 8
Describe and carry out translations, using column vectors
Video lesson: Translations with vectorsComing soon
A robot moves across a grid. It slides along without turning: 3 squares to the right and 2 squares down. Priya describes the move with a vector so that a computer can follow the instructions.

A translation slides every point of a shape by the same amount. It does not turn or flip the shape.

A column vector (x, y) describes the move: the top number is the movement across (positive to the right), and the bottom number is the movement up (positive) or down (negative).

-101234567-2-101234ABCA'B'C'

Key ideas

  • A translation slides a shape; it does not turn or flip it.
  • Column vector: (across, up). Negative numbers mean left or down.
  • Every point moves by the same vector.
  • To find a vector: subtract the original point from the image point.
Worked example 11.1.1
Translate triangle ABC, with A(1, 1), B(3, 1) and C(1, 3), by the vector (3, −2).

Method. Apply the rule to each corner of the shape, one at a time.

  1. Write down the rule. Add the x movement and the y movement to each point.
    Translation by (3, −2)
  2. Find the image of A. Apply the rule to (1, 1).
    A(1, 1) → A′(4, −1)
  3. Find the image of B. Apply the rule to (3, 1).
    B(3, 1) → B′(6, −1)
  4. Find the image of C. Apply the rule to (1, 3).
    C(1, 3) → C′(4, 1)
  5. Write the answer.
    A′(4, −1), B′(6, −1), C′(4, 1)

Check. Each point moves 3 right and 2 down.

Watch out. Check each image is the right distance from the mirror line, or the centre, as the rule requires.

Worked example 11.1.2
Find the vector that moves A(1, 1) to A′(4, −1).

Method. A translation vector shows how far to move in x (top) and in y (bottom).

  1. Find the change in x. Subtract the x values: 4 − (1).
    4 − (1) = 3
  2. Find the change in y. Subtract the y values: −1 − (1).
    −1 − (1) = −2
  3. Write the vector. Put the x movement on top and the y movement below.
    (3, −2)

Check. Check: (1, 1) plus the vector is (4, −1).

Worked example 11.1.3
Translate the point (−2, 5) by the vector (4, −3).

Method. Add the vector to the point, coordinate by coordinate.

  1. Write the point and the vector. Point (−2, 5), vector (4, −3).
    (−2, 5) + (4, −3)
  2. Add the x values. −2 + 4 = 2.
    2
  3. Add the y values. 5 + (−3) = 2.
    2
  4. Write the answer.
    (2, 2)

Check. Move back by (−4, 3) to return to (−2, 5).

Watch out

Watch out: the vector (3, −2) moves 2 DOWN, not up. The bottom number is the movement up or down.

Exam tip

Exam tip. Write the vector in brackets with the x part on top. Check one point by moving it and seeing if it lands where you expect.

Exercise 11A  Translations

Non-calculator
1
FTranslate triangle ABC, with A(1, 1), B(3, 1) and C(1, 3), by the vector (3, −2). Give the new coordinates.
2
FTranslate triangle ABC, with A(1, 1), B(3, 1) and C(1, 3), by the vector (−2, 4). Give the new coordinates.
3
FTranslate triangle ABC, with A(1, 1), B(3, 1) and C(1, 3), by the vector (5, 0). Give the new coordinates.
4
FTranslate triangle ABC, with A(1, 1), B(3, 1) and C(1, 3), by the vector (0, −3). Give the new coordinates.
5
FTranslate triangle ABC, with A(1, 1), B(3, 1) and C(1, 3), by the vector (−4, −1). Give the new coordinates.
6
FTranslate triangle ABC, with A(1, 1), B(3, 1) and C(1, 3), by the vector (2, 3). Give the new coordinates.
7
FTranslate triangle ABC, with A(1, 1), B(3, 1) and C(1, 3), by the vector (−3, 5). Give the new coordinates.
8
FTranslate triangle ABC, with A(1, 1), B(3, 1) and C(1, 3), by the vector (1, −6). Give the new coordinates.
9
FFind the vector that moves (1, 1) to (4, −1).
10
FFind the vector that moves (2, 5) to (−1, 6).
11
FFind the vector that moves (0, 0) to (−3, 2).
12
FFind the vector that moves (−2, 3) to (1, −1).
13
PSThe point (2, 5) is translated by (−3, 1). Find the image.
14
PSA shape is moved 4 squares left and 2 squares up. Write the vector of the movement.
15
RAli says that a translation by (2, 3) and then by (−2, −3) leaves every point where it started. Is Ali right? Explain.
16
RTwo points are A(1, 1) and A′(−2, 4). Are they related by a translation of (−3, 3)? Explain.
17
PSA translation maps the point (1, 2) to (4, 6). Find the vector. Then find the image of (0, 0).
18
FDescribe the translation that moves a shape 5 squares right and 3 squares down, using a column vector.
19
RExplain why translating a shape does not change its size or its shape. Accept any answer that says the points move the same amount, so the shape is unchanged.
11.2

Reflections

Year 8
Reflect a shape in the lines x = a, y = b, y = x and y = −x
Video lesson: ReflectionsComing soon
A mirror is placed on a grid to show how a logo would look from the other side. Tom finds the image of each point by reflecting it in the mirror line.

A reflection flips a shape over a mirror line. Each point ends up the same distance from the mirror line, on the other side.

Key ideas

  • Reflection in x = a: the x value changes; the y value stays.
  • Reflection in y = b: the y value changes; the x value stays.
  • Reflection in y = x: swap the x and y values.
  • Each point is the same distance from the mirror line as its image.
Worked example 11.2.1
Reflect triangle ABC in the line x = 2.

Method. Apply the rule to each corner of the shape, one at a time.

  1. Write down the rule. Use the mirror line x = 2: the x value of each point changes.
    Reflect in x = 2: (x, y) → (4 − x, y)
  2. Find the image of A. Apply the rule to (1, 1).
    A(1, 1) → A′(3, 1)
  3. Find the image of B. Apply the rule to (3, 1).
    B(3, 1) → B′(1, 1)
  4. Find the image of C. Apply the rule to (1, 3).
    C(1, 3) → C′(3, 3)
  5. Write the answer.
    A′(3, 1), B′(1, 1), C′(3, 3)

Check. Each image is the same distance from x = 2 as the original.

Watch out. Check each image is the right distance from the mirror line, or the centre, as the rule requires.

Worked example 11.2.2
Reflect the point (1, 3) in the line y = x.

Method. Swap the x and y values.

  1. Write the rule. For y = x, the x and y values swap.
    (x, y) → (y, x)
  2. Apply the rule. Swap 1 and 3.
    (3, 1)
  3. Write the answer.
    (3, 1)

Check. Reflecting (3, 1) in y = x gives back (1, 3).

Worked example 11.2.3
Reflect the point (2, 1) in the line y = −x.

Method. For y = −x, each point becomes (−y, −x).

  1. Write the rule. For y = −x: (x, y) → (−y, −x).
    (−y, −x)
  2. Apply the rule. Here x = 2 and y = 1, so −y = −1 and −x = −2.
    (−1, −2)
  3. Write the answer.
    (−1, −2)

Check. The midpoint of (2, 1) and (−1, −2) is on y = −x.

Watch out

Watch out: in x = a, the y value does not change. A reflection in x = 2 moves points to the left or right, not up or down.

Exam tip

Exam tip. Count the squares from each point to the mirror line. The image must be the same number of squares on the other side.

Exercise 11B  Reflections

Non-calculator
1
FReflect triangle ABC, with A(1, 1), B(3, 1) and C(1, 3), in the line x = 2.
2
FReflect triangle ABC, with A(1, 1), B(3, 1) and C(1, 3), in the line x = 1.
3
FReflect triangle ABC, with A(1, 1), B(3, 1) and C(1, 3), in the line x = −1.
4
FReflect triangle ABC, with A(1, 1), B(3, 1) and C(1, 3), in the line x = 3.
5
FReflect triangle ABC in the line y = −1.
6
FReflect triangle ABC in the line y = 2.
7
FReflect triangle ABC in the line y = x.
8
FReflect triangle ABC in the line y = −x.
9
FReflect the point (2, 1) in the line y = x. Give the image.
10
FReflect the point (3, −2) in the line x = 0. Give the image.
11
PSThe point P(4, 2) is reflected in x = 1. Find the image, P′.
12
RAli says that reflecting (2, 5) in y = 1 gives (2, −3). Is Ali right? Show your working.
13
PSA point is reflected in y = x, and the image is (−3, 5). What was the original point?
14
RExplain why a point on the mirror line y = x does not move when reflected in it. Accept any answer that says the point is already the same distance from the line on both sides.
11.3

Rotations

Year 8
Rotate a shape about the origin, and about a given point, through 90°, 180° or 270°
Video lesson: RotationsComing soon
A wheel on a toy car turns about its centre. Ben draws the wheel in two positions on a grid, so that he can describe the turn for the instructions.

A rotation turns a shape about a centre through an angle, in a direction (clockwise or anticlockwise).

Key ideas

  • Rotation needs three things: a centre, an angle and a direction.
  • 90° clockwise about the origin: (x, y) → (y, −x).
  • 180° about the origin: (x, y) → (−x, −y).
  • The centre of rotation does not move.
Worked example 11.3.1
Rotate triangle ABC by 90° clockwise about the origin.

Method. Apply the rule to each corner of the shape, one at a time.

  1. Write down the rule. Use the rule for a clockwise quarter turn.
    90° clockwise: (x, y) → (y, −x)
  2. Find the image of A. Apply the rule to (1, 1).
    A(1, 1) → A′(1, −1)
  3. Find the image of B. Apply the rule to (3, 1).
    B(3, 1) → B′(1, −3)
  4. Find the image of C. Apply the rule to (1, 3).
    C(1, 3) → C′(3, −1)
  5. Write the answer.
    A′(1, −1), B′(1, −3), C′(3, −1)

Check. Each image is the same distance from the origin as the original.

Watch out. Check each image is the right distance from the mirror line, or the centre, as the rule requires.

Worked example 11.3.2
Rotate the point (2, 1) by 180° about the origin.

Method. A half turn about the origin changes the sign of both coordinates.

  1. Write the rule. 180°: (x, y) → (−x, −y).
    (−x, −y)
  2. Apply the rule. x = 2 becomes −2, and y = 1 becomes −1.
    (−2, −1)
  3. Write the answer.
    (−2, −1)

Check. A half turn returns the point to the opposite side of the origin.

Worked example 11.3.3
Rotate the point (3, 2) by 90° anticlockwise about the origin.

Method. An anticlockwise quarter turn: (x, y) → (−y, x).

  1. Write the rule. 90° anticlockwise: (x, y) → (−y, x).
    (−y, x)
  2. Apply the rule. y = 2 becomes −2, and x = 3 stays as the second value.
    (−2, 3)
  3. Write the answer.
    (−2, 3)

Check. Rotating back 90° clockwise gives (3, 2) again.

Watch out

Watch out: a clockwise quarter turn is not the same as an anticlockwise quarter turn. Check the direction carefully.

Exam tip

Exam tip. Trace the shape on tracing paper, turn the paper, and check the image lands in the right place.

Exercise 11C  Rotations

Non-calculator
1
FRotate triangle ABC by 90° clockwise about the origin.
2
FRotate triangle ABC by 90° anticlockwise about the origin.
3
FRotate triangle ABC by 180° about the origin.
4
FRotate the point (2, 1) by 180° about the origin.
5
FRotate the point (3, 2) by 90° clockwise about the origin.
6
FRotate the point (4, −1) by 90° anticlockwise about the origin.
7
PSRotate the point (1, 1) by 90° clockwise about the point (1, 1). Give the image.
8
PSRotate the point (3, 1) by 90° clockwise about the point (1, 1). Give the image.
9
RAli says that a rotation of 270° clockwise is the same as 90° anticlockwise. Is Ali right? Explain.
10
PSDescribe the rotation that takes the point (2, 1) to (−1, 2). Give the angle, direction and centre.
11
PSA point is rotated 180° about the origin, and the image is (−4, 6). What was the original point?
12
RExplain why a rotation of 90° clockwise and then 90° clockwise is the same as a rotation of 180°. Accept any answer that adds the two turns.
11.4

Enlargements

Year 8
Enlarge a shape from a centre by a scale factor, Find a scale factor from two lengths
Video lesson: EnlargementsComing soon
A photograph is enlarged from a small print. Each length is multiplied by the same number, the scale factor, so the shape stays the same. Mia draws the triangle at scale factor 2 from the origin.

An enlargement needs a centre and a scale factor k. Each point moves so that its distance from the centre is multiplied by k.

From the origin (0, 0), enlarge by k: (x, y) becomes (kx, ky).

To find the scale factor from two lengths: scale factor = image length ÷ original length.

-101234567-101234567ABCA'B'C'

Key ideas

  • An enlargement needs a centre and a scale factor k.
  • From the origin: each coordinate is multiplied by k.
  • Scale factor = image length ÷ original length.
  • A scale factor between 0 and 1 makes the shape smaller.
Worked example 11.4.1
Enlarge triangle ABC by scale factor 2 from the origin.

Method. Apply the rule to each corner of the shape, one at a time.

  1. Write down the rule. Multiply each coordinate by 2.
    Enlarge from (0, 0) by k = 2: (x, y) → (2x, 2y)
  2. Find the image of A. Apply the rule to (1, 1).
    A(1, 1) → A′(2, 2)
  3. Find the image of B. Apply the rule to (3, 1).
    B(3, 1) → B′(6, 2)
  4. Find the image of C. Apply the rule to (1, 3).
    C(1, 3) → C′(2, 6)
  5. Write the answer.
    A′(2, 2), B′(6, 2), C′(2, 6)

Check. Each image is twice as far from the origin.

Watch out. Check each image is the right distance from the mirror line, or the centre, as the rule requires.

Worked example 11.4.2
The line AB is 2 cm long, and its image A′B′ is 6 cm long. Find the scale factor.

Method. The scale factor is the length of the image divided by the length of the original.

  1. Write the lengths. Original length 2, image length 6.
    6 and 2
  2. Divide the image length by the original. This gives the scale factor.
    6 ÷ 2 = 3
  3. Write the answer.
    3

Check. 2 × 3 = 6.

Worked example 11.4.3
Enlarge the point (3, 1) by scale factor 2, from the centre (1, 1). Find the image.

Method. Find the distance from the centre, multiply by 2, then add the centre back.

  1. Find the distance from the centre. (3, 1) − (1, 1) = (2, 0).
    (2, 0)
  2. Multiply by the scale factor. 2 × (2, 0) = (4, 0).
    (4, 0)
  3. Add the centre back. (4, 0) + (1, 1) = (5, 1).
    (5, 1)
  4. Write the answer.
    (5, 1)

Check. The image is 4 units from (1, 1), which is twice the original distance of 2.

Watch out

Watch out: the scale factor is applied to the distance from the centre, not to the coordinates of the point, unless the centre is the origin.

Exam tip

Exam tip. Write the scale factor first, then the centre. For a scale factor of 1/2, the shape gets smaller.

Exercise 11D  Enlargements

Non-calculator
1
FEnlarge triangle ABC by scale factor 2 from the origin.
2
FEnlarge triangle ABC by scale factor 3 from the origin.
3
FEnlarge triangle ABC by scale factor 1/2 from the origin.
4
FEnlarge triangle ABC by scale factor 2 from the centre (1, 1).
5
FThe length 2 cm is enlarged to 6 cm. Find the scale factor.
6
FThe length 4 cm is enlarged to 10 cm. Find the scale factor.
7
FThe length 3 cm is enlarged to 9 cm. Find the scale factor.
8
FThe length 5 cm is enlarged to 15 cm. Find the scale factor.
9
FThe length 2 cm is enlarged to 1 cm. Find the scale factor.
10
FEnlarge the point (2, 3) by scale factor 3 from the origin. Give the image.
11
PSEnlarge the point (5, 1) by scale factor 2, from the centre (1, 1). Give the image.
12
PSA rectangle is 4 cm long and 3 cm wide. It is enlarged by scale factor 2. What is the new length, in cm?
13
RA triangle has a side 5 cm. The image of the same side is 2 cm. Is the scale factor bigger or smaller than 1? Explain.
14
RAli says that enlarging a shape by scale factor 3 makes its area 3 times bigger. Is Ali right? Explain.
15
PSFind the scale factor for an enlargement that makes a 6 cm side into a 15 cm side.
11.5

Similar and congruent shapes

Year 8
Identify congruent shapes, Use scale factors to find lengths in similar shapes
Video lesson: Similar and congruent shapesComing soon
A small photo is 4 cm wide and 6 cm high. A larger print is 10 cm wide. Sam wants to know the height of the larger print. The two photos are similar, so the same scale factor applies to both sides.

Congruent shapes are exactly the same size and shape. Similar shapes have the same shape, but may be different sizes. All the lengths of similar shapes are in the same ratio, the scale factor.

To find a missing length: find the scale factor (new ÷ old), then multiply the other length by it.

Key ideas

  • Congruent: same size and same shape.
  • Similar: same shape, all lengths in the same ratio (the scale factor).
  • Scale factor = new length ÷ old length.
  • To find a missing length, multiply the other length by the scale factor.
Worked example 11.5.1
Two similar rectangles have widths 4 cm and 10 cm. The first is 6 cm high. Find the height of the second.

Method. Similar shapes have the same ratio for every pair of corresponding lengths. Find the scale factor first.

  1. Find the scale factor. The scale factor is the new length divided by the old length: 10 ÷ 4.
    k = 10 ÷ 4 = 5/2
  2. Multiply the other length by the scale factor. Multiply 6 by 5/2.
    6 × 5/2 = 15
  3. Write the answer.
    15

Check. Check: 15 ÷ 6 = 5/2, the same scale factor.

Worked example 11.5.2
Two similar triangles have a side 3 cm and corresponding side 4.5 cm. Find the scale factor.

Method. The scale factor is the length of the image divided by the length of the original.

  1. Write the lengths. Original length 3, image length 9/2.
    9/2 and 3
  2. Divide the image length by the original. This gives the scale factor.
    9/2 ÷ 3 = 3/2
  3. Write the answer.
    3/2

Check. 3 × 3/2 = 9/2.

Worked example 11.5.3
Triangles with sides 3 cm, 4 cm and 5 cm, and 5 cm, 3 cm and 4 cm, are they congruent?

Method. Congruent shapes have the same lengths, so compare the sides.

  1. List the sides. The first triangle has sides 3, 4, 5. The second has sides 5, 3, 4.
    3, 4, 5 and 5, 3, 4
  2. Compare. The sides are the same lengths, just in a different order.
    3 = 3, 4 = 4, 5 = 5
  3. Write the answer.
    Yes, they are congruent

Check. Congruent triangles have the same three side lengths.

Watch out

Watch out: similar shapes do not have the same lengths. Check the scale factor is the same for every pair of corresponding sides.

Exam tip

Exam tip. Write the ratio of the corresponding sides as a fraction, and simplify. Use the same fraction for every side.

Exercise 11E  Similar and congruent shapes

Non-calculator
1
FTwo similar shapes have corresponding sides 4 cm and 10 cm. A side of the first shape is 6 cm. Find the corresponding side of the second shape.
2
FTwo similar shapes have corresponding sides 2 cm and 5 cm. A side of the first shape is 3 cm. Find the corresponding side of the second shape.
3
FTwo similar shapes have corresponding sides 3 cm and 9 cm. A side of the first shape is 4 cm. Find the corresponding side of the second shape.
4
FTwo similar shapes have corresponding sides 5 cm and 15 cm. A side of the first shape is 7 cm. Find the corresponding side of the second shape.
5
FTwo similar shapes have corresponding sides 4 cm and 6 cm. A side of the first shape is 3 cm. Find the corresponding side of the second shape.
6
FTwo similar shapes have corresponding sides 2 cm and 8 cm. A side of the first shape is 5 cm. Find the corresponding side of the second shape.
7
PSA photo is 4 cm wide and 6 cm high. It is enlarged to 10 cm wide. Find the height of the enlarged photo.
8
PSTwo triangles have sides 3 cm, 4 cm and 5 cm, and sides 6 cm, 8 cm and 10 cm. Are they similar? Find the scale factor.
9
RTwo rectangles are 2 cm by 3 cm, and 4 cm by 6 cm. Are they similar? Explain.
10
RTwo rectangles are 2 cm by 3 cm, and 4 cm by 5 cm. Are they similar? Explain.
11
RTwo triangles have the same angles, but one has sides 3 cm, 4 cm, 5 cm and the other 6 cm, 8 cm, 10 cm. Are they congruent? Explain.
12
PSA model boat is built to a scale of 1 : 20. The model is 30 cm long. How long is the real boat? Give your answer in metres.
13
PSTwo similar triangles have a scale factor of 2.5. A side of the first triangle is 4 cm. Find the corresponding side of the second.
14
RExplain why all squares are similar to each other. Accept any answer that says all squares have the same shape and all their sides are in the same ratio.
15
PSA triangle has sides 2 cm, 3 cm and 4 cm. A similar triangle has its shortest side 6 cm. Find its longest side.

Chapter summary

SkillMethod
TranslationSlide by a column vector (across, up): add the vector to each point
Reflectionx = a: (2a − x, y); y = b: (x, 2b − y); y = x: (y, x); y = −x: (−y, −x)
Rotation about the origin90° clockwise (y, −x); 90° anticlockwise (−y, x); 180° (−x, −y)
EnlargementCentre and scale factor k; from the origin, multiply each coordinate by k
Scale factorImage length ÷ original length
Congruent and similarCongruent: same size and shape. Similar: same shape, lengths in the same ratio

Can you…?

  • I can translate a shape using a column vector, and find a vector between two points
  • I can reflect in x = a, y = b, y = x and y = −x
  • I can rotate a shape about the origin by 90°, 180° and 270°
  • I can enlarge a shape from a centre by a scale factor
  • I can find a scale factor from two lengths, and a missing length in similar shapes
  • I can say whether shapes are congruent, similar or neither

Key terms

translation
a slide of a shape; every point moves by the same vector
vector
a movement described by a column (across, up)
reflection
a flip over a mirror line
rotation
a turn about a centre, through an angle, in a direction
enlargement
a change of size from a centre by a scale factor
scale factor
the number that lengths are multiplied by
congruent
exactly the same size and shape
similar
the same shape, with lengths in the same ratio

Exercise 11Z  Check-up

Non-calculator
1
Translate the point (−1, 4) by the vector (3, −2).
2
Reflect the point (2, −3) in the line x = 0. Give the image.
3
Reflect the point (4, 1) in the line y = x. Give the image.
4
Rotate the point (2, 3) by 180° about the origin. Give the image.
5
Enlarge the point (2, 4) by scale factor 3 from the origin. Give the image.
6
A length of 5 cm is enlarged to 15 cm. Find the scale factor.
7
Two similar shapes: a side 2 cm corresponds to 7 cm. A side of the first shape is 6 cm. Find the corresponding side of the second.
8
Find the vector that moves (1, 1) to (5, −2).
9
Enlarge (1, 2) by scale factor 2 from the origin, then reflect in the line y = x. Give the final point.

Total 25 marks. Allow about 25 minutes.

Exercise 11X  Exam-style questions

Non-calculator
1
FTranslate the point (−2, 3) by the vector (5, −4). Give the image.[2 marks]
2
FReflect the triangle with vertices (1, 1), (3, 1) and (1, 3) in the line x = 0. Give the image of (3, 1).[2 marks]
3
FRotate the point (4, 2) by 90° clockwise about the origin. Give the image.[3 marks]
4
FEnlarge the point (3, 2) by scale factor 2 from the origin. Then find the distance from the origin of the image.[3 marks]
5
FA triangle has sides 4 cm, 6 cm and 8 cm. A similar triangle has a side of 10 cm corresponding to the 4 cm side. Find the other two sides.[4 marks]
6
FFind the scale factor that takes a rectangle 3 cm by 5 cm to one that is 12 cm by 20 cm.[2 marks]
7
FTwo similar rectangles have widths 3 cm and 9 cm. The first rectangle is 4 cm high. Find the height of the second.[3 marks]
8
FExplain how you can tell that two shapes are congruent, and not just similar. Accept any answer that says the sides are the same length, not just in the same ratio.[3 marks]
9
FDescribe fully the single transformation that takes triangle ABC to A′B′C′, when A(1, 1) maps to A′(−1, 1) and B(3, 1) maps to B′(−3, 1).[3 marks]
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