Transformations

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수업 노트

Translations

A translation moves a shape without changing its size or orientation; object and image are congruent.
Movement is described by a column vector \begin{pmatrix} x \\ y \end{pmatrix}: x = horizontal shift (right positive, left negative), y = vertical shift (up positive, down negative).
To translate a shape, move each vertex by the vector and join the new vertices.
To describe a translation, state it is a translation and give the vector (e.g., \begin{pmatrix} 2 \\ -3 \end{pmatrix}).
To reverse a translation, use the same vector with both signs changed.

The object shape and its image after a translation

Reflections

A reflection flips a shape across a mirror line (line of reflection); object and image are congruent.
Each point and its image are equidistant from the mirror line, measured perpendicularly.
Points on the mirror line are invariant (do not move).
Common mirror lines: vertical x = k, horizontal y = k, diagonal y = x or y = -x.
To describe a reflection, state it is a reflection and give the equation of the mirror line.
To reverse a reflection, apply the same reflection again.

Reflecting a shape in a mirror line

Rotations

A rotation turns a shape about a fixed centre of rotation; object and image are congruent.
You must specify the centre, angle (90°, 180°, 270°) and direction (clockwise or anticlockwise). For 180° direction is not needed.
Use tracing paper: draw the shape, place pencil on centre, rotate by the given angle, then draw the image.
To find the centre of rotation for 90° or 270°, use trial and error with tracing paper. For 180°, draw lines connecting corresponding vertices; they intersect at the centre.
To reverse a rotation, rotate by the same angle in the opposite direction about the same centre.

Clockwise and anticlockwise rotation

Enlargements

An enlargement changes the size of a shape by a scale factor (SF) about a centre of enlargement (CoE).
If SF > 1, image is larger; if 0 < SF < 1, image is smaller (fractional enlargement).
For a positive SF, measure horizontal and vertical distances from CoE to a vertex, multiply by SF, and count from CoE to find the image vertex.
For a negative SF, the image is rotated 180° and lies on the opposite side of the CoE; distances are multiplied by |SF|.
To describe an enlargement, state it is an enlargement, give the SF and the coordinates of the CoE.
To reverse an enlargement, use the reciprocal SF with the same CoE (sign stays for negative SF).
Area scale factor = (linear SF)²; multiply original area by this to find image area.

Describing an enlargement

Combined Transformations

When two or more transformations are applied sequentially, the final image can often be described by a single transformation.
To find the single transformation, consider the net effect on position, orientation and size.
Common combinations: rotation + translation can sometimes be a single rotation about a different centre; two reflections can be a rotation or translation.

슬라이드

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연습 문제

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1.What is the name of the transformation that moves a shape without changing its size or orientation?
Easy
- ATranslation
- BReflection
- CRotation
- DEnlargement
2.A shape is translated by the vector \begin{pmatrix} -3 \\ 5 \end{pmatrix}. What does this mean?
Easy
- A3 left and 5 up
- B3 right and 5 down
- C3 left and 5 down
- D3 right and 5 up
3.What is the equation of the mirror line for a reflection that maps a point (2, 3) to (-2, 3)?
Easy
- Ax = 0
- By = 0
- Cx = 2
- Dy = 3
4.A rotation turns a shape. Which of the following is NOT needed to fully describe a rotation?
Easy
- AScale factor
- BCentre of rotation
- CAngle of rotation
- DDirection of rotation
5.An enlargement has scale factor 0.5. What does this do to the size of the shape?
Easy
- AMakes it smaller
- BMakes it larger
- CKeeps it the same size
- DFlips it over
6.When describing a translation, what must you include?
Easy
- AA column vector
- BA mirror line
- CA centre of rotation
- DA scale factor
7.Triangle A has vertices (1, 2), (3, 2), (1, 5). It is translated by vector \begin{pmatrix} -2 \\ 4 \end{pmatrix}. What are the coordinates of the image of vertex (1, 2)?
Medium
- A(-1, 6)
- B(3, -2)
- C(-1, -2)
- D(3, 6)
8.A shape is reflected in the line y = 2. Which of the following points is invariant?
Medium
- A(3, 2)
- B(2, 3)
- C(0, 0)
- D(1, 1)

더 많은 문제

9.A triangle is rotated 90° clockwise about the origin. The image of point (2, 1) is:
Medium
- A(1, -2)
- B(-1, 2)
- C(-2, -1)
- D(2, -1)
10.A shape is enlarged with scale factor 3 and centre (0, 0). The point (2, 1) maps to:
Easy
- A(6, 3)
- B(2/3, 1/3)
- C(5, 2)
- D(6, 1)
11.Describe fully the transformation that maps shape A to shape B if shape B is a 180° rotation of shape A about (0, 0).
Easy
- ARotation 180° about (0, 0)
- BReflection in the x-axis
- CTranslation by vector (0, 0)
- DEnlargement scale factor -2 about (0, 0)
12.A shape is reflected in the line y = x. The point (2, 5) maps to:
Easy
- A(5, 2)
- B(-2, -5)
- C(-5, -2)
- D(2, -5)
13.A triangle is enlarged by scale factor -2 with centre (1, 1). The point (3, 4) on the original maps to:
Medium
- A(-3, -5)
- B(5, 7)
- C(-1, -3)
- D(7, 5)
14.A shape is enlarged by scale factor 1/3 with centre (0, 0). The image has area 4 cm². What is the area of the original shape?
Medium
- A36 cm²
- B12 cm²
- C4/9 cm²
- D4 cm²
15.Triangle A is rotated 90° anticlockwise about (2, 1) to give triangle B. Then B is translated by vector \begin{pmatrix} -3 \\ 2 \end{pmatrix} to give triangle C. Describe a single transformation that maps A to C.
Hard
- ARotation 90° anticlockwise about (-0.5, 0.5)
- BRotation 90° anticlockwise about (-1, 3)
- CReflection in line x = -1
- DRotation 90° clockwise about (2, 1)
16.A shape is reflected in the line x = 1 and then reflected in the line y = -1. The combined transformation is equivalent to:
Hard
- AA rotation of 180° about (1, -1)
- BA translation by vector (2, -2)
- CA reflection in the line y = x
- DA rotation of 90° about (0, 0)
17.A translation moves a shape by the vector \begin{pmatrix} 5 \\ -2 \end{pmatrix}. What does this mean?
Easy
- A5 units right and 2 units down
- B5 units left and 2 units up
- C5 units right and 2 units up
- D5 units left and 2 units down
18.A shape is reflected in the line y = -1. What does this mean?
Easy
- AThe mirror line is a horizontal line passing through -1 on the y-axis
- BThe mirror line is a vertical line passing through -1 on the x-axis
- CThe mirror line is the diagonal line y = -x
- DThe mirror line is the horizontal line y = 1
19.A shape is rotated 90° clockwise about the origin. The image of point (3, 1) is:
Medium
- A(1, -3)
- B(-1, 3)
- C(-3, -1)
- D(1, 3)
20.An enlargement has scale factor 2. What does this do to the size of the shape?
Easy
- AThe shape becomes twice as large in each dimension
- BThe shape becomes half as large in each dimension
- CThe shape becomes twice as large in area only
- DThe shape stays the same size
21.Triangle A has vertices (1, 1), (4, 1), (4, 3). It is reflected in the line x = 0. What are the coordinates of the image of vertex (4, 3)?
Easy
- A(-4, 3)
- B(4, -3)
- C(-4, -3)
- D(0, 3)
22.A shape is enlarged with scale factor \frac{1}{2} and centre (0, 0). The point (6, -4) maps to:
Easy
- A(3, -2)
- B(12, -8)
- C(-3, 2)
- D(6, -4)
23.Describe fully the single transformation that maps triangle A to triangle B if triangle B is a translation of triangle A by the vector \begin{pmatrix} -3 \\ 4 \end{pmatrix}.
Easy
- ATranslation by \begin{pmatrix} -3 \\ 4 \end{pmatrix}
- BTranslation by \begin{pmatrix} 3 \\ -4 \end{pmatrix}
- CReflection in the line y = x
- DRotation 90° anticlockwise about the origin
24.A shape is reflected in the line y = x. The point (4, -1) maps to:
Easy
- A(-1, 4)
- B(1, -4)
- C(-4, 1)
- D(4, -1)
25.A triangle is enlarged by scale factor 3 with centre (2, 1). The point (4, 2) on the original maps to:
Medium
- A(8, 4)
- B(6, 3)
- C(10, 5)
- D(4, 2)
26.What is the name of the transformation that flips a shape across a line?
Easy
- AReflection
- BRotation
- CTranslation
- DEnlargement
27.A shape is reflected in the line y = -1. What does this mean?
Easy
- AEach point is reflected across the horizontal line y = -1
- BEach point is reflected across the vertical line y = -1
- CThe shape is rotated about the point (0, -1)
- DThe shape is translated down by 1 unit
28.Triangle A has vertices (1, 2), (3, 2), (1, 5). It is translated by vector \begin{pmatrix} -2 \\ 4 \end{pmatrix}. What are the coordinates of the image of vertex (1, 5)?
Easy
- A(-1, 9)
- B(-1, 1)
- C(3, 9)
- D(-2, 4)
29.What is the name of the transformation that changes the size of a shape but not its orientation?
Easy
- ATranslation
- BReflection
- CRotation
- DEnlargement
30.A translation moves a shape without changing its size or orientation.
Easy
True or false?
31.A shape is reflected in the line y = x. The point (3, -2) maps to which point?
Easy
- A(-3, 2)
- B(-2, 3)
- C(3, 2)
- D(2, -3)
32.Which of the following are needed to fully describe a rotation? (Select all that apply)
Medium
- ACentre of rotation
- BAngle of rotation
- CDirection of rotation
- DScale factor
- EMirror line
33.Match each transformation with its effect on the shape.
Easy
- Translation
- Reflection
- Rotation
- Enlargement
- Changes position only
- Flips across a line
- Turns about a point
- Changes size
34.Place the following steps for enlarging a shape in the correct order.
Medium
- Multiply distances by scale factor
- Count horizontal and vertical distances from centre of enlargement to a vertex
- Mark the centre of enlargement
- Draw the enlarged image
35.A rotation of 90° clockwise about the origin maps the point (2, 3) to (-3, 2).
Medium
True or false?
36.A translation is described by a scale factor and a centre of enlargement.
Easy
True or false?
37.Which of the following are needed to fully describe a reflection? (Select all that apply)
Easy
- AThe equation of the mirror line
- BThe centre of rotation
- CThat the transformation is a reflection
- DThe scale factor
- EThe direction of rotation
38.A triangle is rotated 180° about the origin. The image of point (4, -1) is (a, b). Find a + b.
Easy
39.Complete the sentence about transformations.
Easy
In a reflection, points on the mirror line are called ____ points.
40.A shape is enlarged with scale factor 1/2. What happens to the area of the shape?
Medium
41.A shape is enlarged with scale factor -2 and centre (1, 1). The point (3, 4) on the original maps to (a, b). Find a + b.
Hard
42.A shape with area 36 cm² is enlarged by scale factor \frac{2}{3}. Find the area of the image in cm².
Medium
43.Describe fully the single transformation that maps triangle A with vertices (1, 1), (3, 1), (3, 3) to triangle B with vertices (2, 2), (4, 2), (4, 4).
Medium
44.A triangle is rotated 180° about the origin. The image of point (4, -1) is (a, b). Find a + b.
Easy
- A-3
- B-1
- C-4
- D-2
45.A shape with area 36 cm² is enlarged by scale factor \frac{2}{3}. Find the area of the image in cm².
Medium
- A17 cm²
- B15 cm²
- C16 cm²
- D18 cm²

복습 노트

## Transformations (IGCSE Maths Extended) ### Key Concepts - **Object**: original shape; **Image**: transformed shape (vertices labelled A', B', etc.) - Four types: **translation**, **reflection**, **rotation**, **enlargement** ### Translations - Moves shape without changing size/orientation; object and image are **congruent**. - Described by a column vector \(\begin{pmatrix} x \\ y \end{pmatrix}\): - \(x\): horizontal (+ right, – left) - \(y\): vertical (+ up, – down) - **To describe**: state "translation" + vector. - **Reverse**: change signs of both components. ### Reflections - Flips shape across a **mirror line** (line of reflection). - Perpendicular distances from mirror line are equal; points on line are **invariant**. - Mirror lines: vertical \(x = k\), horizontal \(y = k\), diagonal \(y = x\) or \(y = -x\). - **To describe**: state "reflection" + equation of mirror line. - **Reverse**: same reflection again. ### Rotations - Turns shape about a **centre of rotation** by a given angle and direction. - Angles: 90°, 180°, 270°; direction: clockwise or anti-clockwise (omit for 180°). - Use tracing paper: draw arrow up, rotate, check orientation. - **To describe**: state "rotation", centre, angle, direction. - **Reverse**: opposite direction, same centre and angle. ### Enlargements - Changes size by **scale factor** (sf): - \(|sf| > 1\): larger - \(0 < |sf| < 1\): smaller - Negative sf: also rotates 180° (image on opposite side of centre). - **Centre of enlargement (CoE)** determines position. - **Area scale factor** = \((\text{length sf})^2\). - **To describe**: state "enlargement", sf, CoE coordinates. - **Reverse**: sf becomes reciprocal (keep sign), same CoE. ### Common Errors - Translation vector is the movement, not the gap between shapes. - For reflection, don't confuse \(x = k\) (vertical) with \(y = k\) (horizontal). - For negative enlargement, count distances on opposite side of CoE. ### Quick Tips - Use tracing paper for rotations and to find centres. - For enlargement, draw lines from CoE through vertices to check. - Always label images clearly (e.g., A', B').

Translation: A transformation that moves every point of a shape by the same distance in the same direction, described by a column vector.
Reflection: A transformation that flips a shape across a line (mirror line), producing a mirror image.
Rotation: A transformation that turns a shape about a fixed point (centre of rotation) by a given angle and direction.
Enlargement: A transformation that changes the size of a shape by a scale factor, with respect to a centre of enlargement.
Scale factor: The multiplier applied to distances from the centre of enlargement; determines size change. Negative values also rotate the shape 180°.
Centre of enlargement: The fixed point from which distances are measured in an enlargement; lines through corresponding vertices meet here.
Invariant point: A point that does not move during a transformation (e.g., points on the mirror line in a reflection, or the centre of rotation).
Object: The original shape before a transformation.
Image: The shape after a transformation; vertices are often labelled with primes (e.g., A').

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