Transformations

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Apuntes de la lección

What a Transformation Is

A transformation changes the position, size or orientation of a shape on a grid.
The original shape is called the object; the shape after the transformation is the image.
Every point of the object moves to a matching point on the image, so the two shapes have the same number of vertices in the same order.
Transformations are described by naming the type and giving the exact details needed to carry it out.
The four transformations you need are reflection, rotation, translation and enlargement.

Reflection

A reflection flips a shape over a straight line called the mirror line.
The image is the same size and shape as the object, but it is a mirror image — the two are congruent.
Each point and its image are the same perpendicular distance from the mirror line, on opposite sides of it.
The mirror line is the perpendicular bisector of the segment joining any point to its image.
To describe a reflection fully, state the equation of the mirror line, for example x = 2, y = -1 or y = x.
Reflecting in the line y = x swaps the coordinates of each point: (x, y) maps to (y, x).
Reflecting in the x-axis maps (x, y) to (x, -y); reflecting in the y-axis maps (x, y) to (-x, y).

Rotation

A rotation turns a shape about a fixed point called the centre of rotation.
A rotation is described by three things: the centre, the angle and the direction (clockwise or anticlockwise).
Angles are usually 90°, 180° or 270°; a 180° turn looks the same clockwise or anticlockwise.
The image is the same size and shape as the object, so object and image are congruent.
Every point and its image are the same distance from the centre of rotation.
To find the centre, join a point to its image and repeat for another point; the perpendicular bisectors meet at the centre.
A rotation of 360° returns a shape to its starting position.

Translation

A translation slides a shape without turning or flipping it.
Every point moves the same distance in the same direction, so the image is congruent to the object.
A translation is described by a column vector written as two numbers in brackets, one above the other.
The top number gives the movement right (positive) or left (negative); the bottom number gives the movement up (positive) or down (negative).
For example, the vector \begin{pmatrix} 3 \\ -2 \end{pmatrix} means 3 right and 2 down.
A translation does not change the orientation of a shape — the image is not turned or reflected.

Enlargement

An enlargement changes the size of a shape from a fixed point called the centre of enlargement.
It is described by the centre and the scale factor.
With a positive scale factor greater than 1 the image is larger; with a scale factor between 0 and 1 the image is smaller.
The image is the same shape as the object, so object and image are similar — corresponding angles are equal and corresponding lengths are in the same ratio.
Each length on the image is the scale factor multiplied by the matching length on the object.
To find the centre, draw straight lines through pairs of corresponding vertices; the lines meet at the centre of enlargement.
The centre may lie inside, on, or outside the object.

Describing a Single Transformation

Always name the type of transformation first: reflection, rotation, translation or enlargement.
For a reflection, give the equation of the mirror line.
For a rotation, give the centre, the angle and the direction.
For a translation, give the column vector.
For an enlargement, give the centre and the scale factor.
Check whether the image is the same size as the object: if it is, the transformation is a reflection, rotation or translation; if not, it is an enlargement.

Congruence and Similarity

Congruent shapes are exactly the same shape and the same size.
Reflections, rotations and translations all produce congruent images.
Similar shapes are the same shape but may be different sizes; corresponding angles are equal and corresponding lengths are in the same ratio.
An enlargement produces a similar image, not a congruent one (unless the scale factor is 1).
Congruence is a special case of similarity where the ratio of corresponding lengths is 1.

Symmetry

A shape has line symmetry if it can be folded along a line so that the two halves match exactly; that line is a line of symmetry (or mirror line).
The number of lines of symmetry of a shape can be counted by testing every possible fold line.
A shape has rotational symmetry if it looks the same after being turned about its centre by less than a full turn.
The order of rotational symmetry is the number of positions in which the shape looks the same during one full 360° turn.
A square has 4 lines of symmetry and rotational symmetry of order 4; a rectangle has 2 lines of symmetry and rotational symmetry of order 2.
An equilateral triangle has 3 lines of symmetry and rotational symmetry of order 3.

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Preguntas de práctica

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1.Which single transformation preserves distances and oriented angles?
Easy
- ATranslation
- BEnlargement
- CShear
- DProjection
2.A shape is enlarged by scale factor 3. Which property of the shape is preserved?
Easy
- AAngles
- BDistances
- CArea
- DPerimeter
3.Which transformation preserves parallelism?
Medium
- AAffine transformation
- BProjective transformation
- CHomeomorphism
- DDiffeomorphism
4.A point P(2, 3) is reflected in the line y = x. What are the coordinates of its image?
Medium
- A(3, 2)
- B(2, 3)
- C(-2, 3)
- D(2, -3)
5.A shape is translated by the column vector \begin{pmatrix} -4 \\ 2 \end{pmatrix}. Which single transformation would return it to its original position?
Medium
- ATranslation by \begin{pmatrix} 4 \\ -2 \end{pmatrix}
- BTranslation by \begin{pmatrix} -4 \\ 2 \end{pmatrix}
- CReflection in the x-axis
- DRotation of 180° about the origin
6.Which of the following transformations preserve angles? (select all that apply)
Medium
- ATranslation
- BRotation
- CEnlargement
- DShear
- EProjection
7.Which of the following are true about a rotation? (select all that apply)
Medium
- AIt preserves distances
- BIt preserves angles
- CIt preserves orientation
- DIt changes the size of the shape
- EIt is a type of isometry
8.A reflection is an isometry.
Easy
True or false?

Más preguntas

9.An enlargement with scale factor 2 preserves area.
Easy
True or false?
10.Match each transformation to the property it preserves.
Medium
- Translation
- Enlargement
- Projective transformation
- Distances and oriented angles
- Angles and ratios between distances
- Collinearity
11.Order these transformations from the most specific (preserving the most properties) to the least specific (preserving the fewest properties):
Medium
- Displacement
- Isometry
- Similarity
- Affine transformation
- Projective transformation
12.Match each transformation to the type of movement it describes.
Medium
- Translation
- Reflection
- Rotation
- Enlargement
- Sliding without turning or flipping
- Flipping over a mirror line
- Turning about a fixed centre
- Resizing by a scale factor
13.Order these steps to describe fully the single transformation that maps a shape onto its image.
Medium
- State the type of transformation
- Give the mirror line, centre and angle, or column vector
- Check the image is congruent or similar to the original
14.A shape is rotated 90° clockwise about the origin. Which of the following are true? (select all that apply)
Medium
- AThe image is congruent to the original
- BThe image is a mirror image of the original
- CThe image is similar to the original
- DThe orientation of the shape is reversed
- EThe image has the same area as the original
15.A point is reflected in the line y = x. Which of the following are true? (select all that apply)
Hard
- AThe x- and y-coordinates swap
- BThe point moves to the opposite side of the line y = x
- CThe distance from the point to the line y = x is unchanged
- DThe point moves to the opposite side of the x-axis
- EThe point moves to the opposite side of the y-axis
16.A point P(3, -2) is reflected in the x-axis. What are the coordinates of its image?
Medium
- A(3, 2)
- B(-3, -2)
- C(-3, 2)
- D(2, -3)
17.A point P(5, 1) is rotated 90° anticlockwise about the origin. What are the coordinates of its image?
Medium
- A(-1, 5)
- B(1, -5)
- C(-5, -1)
- D(1, 5)
18.A point P(4, 7) is translated by the column vector \begin{pmatrix} -2 \\ 3 \end{pmatrix}. What are the coordinates of its image?
Medium
- A(2, 10)
- B(6, 4)
- C(2, 4)
- D(6, 10)
19.A shape is enlarged by scale factor 2 about the origin. A vertex of the shape is at (3, -1). What are the coordinates of its image?
Hard
- A(6, -2)
- B(1.5, -0.5)
- C(6, 2)
- D(5, 1)
20.A rotation about a centre preserves distances and angles.
Easy
True or false?
21.Describe fully the single transformation that maps the point (3, 5) to the point (3, -5).
Medium
- AReflection in the x-axis
- BReflection in the y-axis
- CRotation of 180° about the origin
- DTranslation by the vector \begin{pmatrix} 0 \\ -5 \end{pmatrix}
22.The point (2, -3) is rotated 180° about the origin. What are the coordinates of its image?
Medium
- A(-2, 3)
- B(2, 3)
- C(-2, -3)
- D(3, -2)
23.Which of the following statements about a reflection are true? (select all that apply)
Medium
- AA reflection is an isometry, so it preserves distances.
- BA reflection reverses the orientation of a shape.
- CA reflection preserves the size of angles.
- DA point on the mirror line stays in the same position.
- EA reflection changes the area of a shape.
24.Match each transformation to the number of lines of symmetry of its image when the object is a rectangle 3 cm by 5 cm.
Medium
- Identity (no change)
- Reflection in a diagonal
- Enlargement by scale factor 2
- 2 lines of symmetry
- 0 lines of symmetry
25.Order these transformations to show the result of applying them one after another to the point (1, 0): first a translation by \begin{pmatrix} 2 \\ 0 \end{pmatrix}, then a reflection in the x-axis, then a rotation of 90° anticlockwise about the origin.
Medium
- (1, 0)
- (3, 0)
- (3, 0)
- (0, 3)
26.An enlargement with a positive scale factor less than 1 produces an image smaller than the object.
Easy
True or false?
27.A shape is translated by the column vector \begin{pmatrix} 5 \\ -2 \end{pmatrix}. Which single transformation would return it to its original position?
Medium
- ATranslation by \begin{pmatrix} -5 \\ 2 \end{pmatrix}
- BTranslation by \begin{pmatrix} 5 \\ 2 \end{pmatrix}
- CTranslation by \begin{pmatrix} -5 \\ -2 \end{pmatrix}
- DTranslation by \begin{pmatrix} 2 \\ -5 \end{pmatrix}
28.A triangle has vertices at A(1, 2), B(3, 2) and C(1, 5). It is enlarged by scale factor 2 about the origin. What is the area of the image triangle?
Hard
- A6 square units
- B12 square units
- C24 square units
- D48 square units
29.A point P(2, -3) is reflected in the line y = x and then translated by the vector \begin{pmatrix} 1 \\ 4 \end{pmatrix}. What are the coordinates of the final image?
Medium
- A(-3, 3)
- B(-3, 5)
- C(3, -3)
- D(-2, 1)
30.A square is rotated 90° clockwise about its centre and then reflected in a vertical line through its centre. Which of the following statements are true? (select all that apply)
Hard
- AThe final image is congruent to the original square.
- BThe final image has the same orientation as the original square.
- CThe final image is a translation of the original square.
- DThe final image has the same area as the original square.
- EThe final image is a reflection of the original square.
31.A shape is enlarged by scale factor 1.5. The perimeter of the image is 1.5 times the perimeter of the original shape.
Medium
True or false?
32.Order these transformations from the one that preserves the most properties to the one that preserves the fewest.
Hard
- Displacement
- Isometry
- Similarity
- Affine transformation
- Projective transformation
33.A shape is transformed by a reflection in the line y = x followed by a rotation of 90° anticlockwise about the origin. The combined transformation is equivalent to a single transformation. Which one?
Challenge
- AReflection in the y-axis
- BReflection in the x-axis
- CRotation of 90° clockwise about the origin
- DReflection in the line y = -x
34.A point P(4, -1) is rotated 180° about the point (1, 2). What are the coordinates of its image?
Medium
- A(-2, 5)
- B(-2, 3)
- C(2, 5)
- D(-2, -5)
35.Which class of transformation preserves distances and oriented angles?
Easy
- Adisplacement
- Bisometry
- Csimilarity
- Daffine transformation
36.Which class of transformation preserves angles and the ratios between distances?
Easy
- Adisplacement
- Bisometry
- Csimilarity
- Dprojective transformation
37.Which class of transformation preserves collinearity?
Easy
- Adisplacement
- Bsimilarity
- Caffine transformation
- Dprojective transformation
38.Which class of transformation preserves parallelism?
Easy
- Adisplacement
- Bisometry
- Caffine transformation
- Dprojective transformation
39.A geometric transformation is described as a bijection of a set to itself. What does this mean about the transformation?
Medium
- AIt maps every point to the same point.
- BIt is one-to-one and onto, so it has an inverse.
- CIt only works in three dimensions.
- DIt must preserve distances.
40.Which of the following are examples of the transformation classes listed in the classification of geometric transformations? (select all that apply)
Medium
- Adisplacements
- Bisometries
- Csimilarities
- Daffine transformations
- EMöbius transformations
- Ftemperature conversions
41.What is the name of the transformation that preserves collinearity?
Medium
42.A shape is enlarged by scale factor 3. By what factor is its area multiplied?
Medium
43.Complete the sentence about transformations.
Easy
A ____ transformation preserves angles and ratios between distances.
44.A rectangle has area 12 cm². It is enlarged by scale factor 3. What is the area of the image, in cm²?
Medium
45.Complete the sentence about an enlargement.
Medium
An enlargement with scale factor k multiplies lengths by ____, so the area is multiplied by ____.
46.A shape is enlarged by scale factor 4. By what number is its area multiplied?
Medium
47.Complete the sentence about the symmetry of a square.
Medium
A square has ____ lines of symmetry and rotational symmetry of order ____.
48.A triangle has an area of 10 cm². It is enlarged by scale factor 3. What is the area of the image, in cm²?
Medium
49.What is the name of the transformation that turns a shape about a fixed point?
Easy
50.Complete the sentence about the hierarchy of transformations.
Medium
Each class of transformation contains the ____ one, so displacements are also isometries.
51.A rectangle has an area of 15 cm². It is enlarged by scale factor 2. What is the area of the image, in cm²?
Medium
52.A triangle has an area of 8 cm². It is enlarged by scale factor 4. What is the area of the image, in cm²?
Hard
53.Describe fully the single transformation that maps a shape onto its image when the image is congruent but rotated 90° clockwise about the origin.
Hard
54.Describe fully the single transformation that maps a shape onto its image when the image is congruent but has been flipped over the line y = x.
Hard
55.Describe fully the single transformation that maps a shape onto its image when the image is congruent but has been flipped over the line y = -x.
Hard
56.Explain the difference between a displacement and an isometry, giving an example of each.
Medium
57.Explain why every displacement is also an isometry, but not every isometry is a displacement.
Hard
58.A shape is enlarged by scale factor 3. By what factor is its area multiplied?
Medium
- A11
- B9
- C8
- D10
59.A rectangle has area 12 cm². It is enlarged by scale factor 3. What is the area of the image, in cm²?
Medium
- A108 cm²
- B97 cm²
- C130 cm²
- D119 cm²
60.A shape is enlarged by scale factor 4. By what number is its area multiplied?
Medium
- A16
- B17
- C15
- D18
61.A triangle has an area of 10 cm². It is enlarged by scale factor 3. What is the area of the image, in cm²?
Medium
- A99 cm²
- B108 cm²
- C90 cm²
- D81 cm²
62.A rectangle has an area of 15 cm². It is enlarged by scale factor 2. What is the area of the image, in cm²?
Medium
- A66 cm²
- B60 cm²
- C72 cm²
- D54 cm²
63.A triangle has an area of 8 cm². It is enlarged by scale factor 4. What is the area of the image, in cm²?
Hard
- A141 cm²
- B154 cm²
- C128 cm²
- D115 cm²

Apuntes de repaso

## Transformations — KS3 Revision A **transformation** moves or changes a shape. The original shape is the **object**; the result is the **image**. The four transformations you need are reflection, rotation, translation and enlargement. ### Reflection - A reflection flips a shape over a **mirror line**. - The image is the same size and shape as the object, but reversed (a mirror image). - Every point and its image are the same perpendicular distance from the mirror line, on opposite sides. - You may be asked to reflect in lines such as the x-axis, the y-axis, or the line y = x. ### Rotation - A rotation turns a shape about a fixed **centre of rotation**. - You must give the **angle** and the **direction** (clockwise or anticlockwise). - The shape keeps the same size and orientation is turned; distances and angles are unchanged. ### Translation - A translation slides a shape without turning or flipping it. - It is described by a **column vector**, written as \(\begin{pmatrix} x \\ y \end{pmatrix}\), where x is the movement right (positive) or left (negative) and y is the movement up (positive) or down (negative). - Every point moves by the same vector, so the image is congruent to the object. ### Enlargement - An enlargement changes the size of a shape from a **centre of enlargement** using a **scale factor**. - For a positive scale factor k, each image point is k times as far from the centre as the object point. - If k > 1 the shape gets bigger; if 0 < k < 1 it gets smaller. - The image is **similar** to the object: angles are equal and corresponding lengths are in the same ratio. ### Describing a single transformation fully - **Reflection:** name the mirror line. - **Rotation:** give the centre, the angle and the direction. - **Translation:** give the column vector. - **Enlargement:** give the centre and the scale factor. ### Congruence and similarity - **Congruent** shapes are identical in shape and size. Reflections, rotations and translations produce congruent images. - **Similar** shapes have equal angles and corresponding sides in the same ratio. Enlargement produces similar images. ### Symmetry - **Line symmetry:** a shape has a line of symmetry if reflecting it in that line leaves it unchanged. A shape can have 0, 1 or many lines of symmetry. - **Rotational symmetry:** the order is the number of times a shape fits onto itself in a full 360° turn. ### Quick check - Reflection → flip; rotation → turn; translation → slide; enlargement → resize. - Always label the image clearly and check the direction of movement.

Transformation: A rule that moves or changes a shape; the original is the object and the result is the image.
Reflection: A transformation that flips a shape over a mirror line; the image is the same size and shape but reversed.
Rotation: A transformation that turns a shape about a centre of rotation by a given angle and direction.
Translation: A transformation that slides a shape without turning or flipping it, described by a column vector.
Column vector: A pair of numbers written vertically, \(\begin{pmatrix} x \\ y \end{pmatrix}\), giving the horizontal and vertical movement in a translation.
Enlargement: A transformation that changes the size of a shape from a centre of enlargement using a scale factor.
Scale factor: The multiplier in an enlargement; for positive k, image distances from the centre are k times the object distances.
Congruent: Identical in shape and size; produced by reflection, rotation or translation.
Similar: Having equal angles and corresponding sides in the same ratio; produced by enlargement.
Line symmetry: A shape has line symmetry if reflecting it in a line leaves it unchanged.
Rotational symmetry: The order is the number of times a shape fits onto itself during a full 360° turn.

Tarjetas

What is a geometric transformation?

A rule that maps every point of a shape to a new point, with some geometrical property preserved, such as distances, angles or ratios.

What is a reflection?

A transformation that flips a shape over a mirror line, producing a mirror image. Distances and angles are preserved, but the orientation is reversed.

Which properties does a reflection preserve?

It preserves distances and angles, so the image is congruent to the object, but it reverses orientation.

How do you reflect a point in the line y = x?

Swap the coordinates: the point (a, b) maps to (b, a).

How do you reflect a point in the x-axis?

The point (a, b) maps to (a, −b): the y-coordinate changes sign.

How do you reflect a point in the y-axis?

The point (a, b) maps to (−a, b): the x-coordinate changes sign.

What is a rotation?

A transformation that turns a shape about a fixed centre by a given angle and direction. Distances and angles are preserved.

What three pieces of information fully describe a rotation?

The centre of rotation, the angle of rotation, and the direction (clockwise or anticlockwise).

What is a translation?

A transformation that slides every point of a shape the same distance in the same direction, without turning or flipping it.

How is a translation described?

By a column vector. The top number is the horizontal movement (right is positive) and the bottom number is the vertical movement (up is positive).

What does the column vector (3, −2) mean as a translation?

Move every point 3 units right and 2 units down.

What is an enlargement?

A transformation that changes the size of a shape by a scale factor from a fixed centre. Angles are preserved but lengths are multiplied by the scale factor.

What two pieces of information fully describe an enlargement?

The centre of enlargement and the scale factor.

How do you find the image of a point under an enlargement?

Measure the distance and direction from the centre to the point, then multiply that distance by the scale factor to locate the image point.

What is the difference between congruence and similarity?

Congruent shapes are identical in shape and size; similar shapes have the same shape but may differ in size, with corresponding lengths in a fixed ratio.

Which transformations always produce a congruent image?

Reflection, rotation and translation. They preserve distances and angles, so the image is congruent to the object.

Which transformation produces a similar but not necessarily congruent image?

Enlargement, unless the scale factor is 1. It preserves angles but multiplies all lengths by the scale factor.

What is line symmetry?

A shape has line symmetry if a mirror line can be drawn so that reflecting the shape in that line maps it onto itself.

What is rotational symmetry?

A shape has rotational symmetry if it maps onto itself when rotated about a centre by an angle smaller than 360°. The order is the number of positions in which it looks the same in a full turn.

How do you find the order of rotational symmetry of a shape?

Count how many times the shape fits onto itself during one full 360° turn about its centre.

What must you state when fully describing a single transformation?

Its type, plus the details: the mirror line for a reflection, the centre, angle and direction for a rotation, the column vector for a translation, or the centre and scale factor for an enlargement.

What happens to a shape when it is transformed by a combination of transformations?

Each transformation is applied in turn to the image produced by the previous one, giving a single combined result.

A shape is reflected in a line and then reflected in a parallel line. What single transformation is equivalent to this?

A translation, since two reflections in parallel lines combine to give a translation.

What is the relationship between the image and object under an isometry?

They are congruent: an isometry preserves both angles and distances.

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