Grade 9 Mathematics
Grade 9 · Term 4Mathematics

Transformation Geometry

We perform and describe reflections, translations, rotations, and enlargements/reductions on the Cartesian plane. We investigate the properties preserved under each transformation and use coordinates to describe transformations algebraically.

11.1 Reflections and Translations11.2 Rotations and Enlargements/Reductions
Weeks 6–7

11.1 Reflections and Translations

  • Reflect figures across the x-axis, y-axis, the line y = x, and y = −x
  • Translate figures by a given vector (horizontal and vertical shift)
  • Describe transformations using coordinate notation
  • Identify properties preserved: shape, size, orientation
🌍

Real-World Connection

Reflections appear in mirror images, bilateral symmetry in nature (butterfly wings, human faces), and architecture (symmetrical buildings). Translations appear in repeating patterns: wallpaper, tiling, and manufacturing conveyors that move objects horizontally. Computer graphics use these transformations millions of times per second to render animations.

Cartesian Planexy-33-22-111-12-23-30Q IQ IIQ IIIQ IV
The Cartesian plane with all four quadrants. Transformations move or resize figures on this plane.

Reflection Rules

Reflection Line

Coordinate Rule

x-axis

(x, y) → (x, −y)

y-coordinate changes sign

y-axis

(x, y) → (−x, y)

x-coordinate changes sign

y = x

(x, y) → (y, x)

x and y coordinates swap

y = −x

(x, y) → (−y, −x)

Swap and negate both

Property / Rule

Translation

A translation moves every point by the same distance in the same direction. Described by a vector (a, b): move a units horizontally and b units vertically.

(x,y)→(x+a,  y+b)(x, y) \to (x+a, \; y+b)

ℹ️ Note

Reflections and translations are ISOMETRIES — they preserve size and shape (congruent image). The orientation is preserved under translation, but REVERSED under reflection (mirror image).

Worked Examples

Worked Example

Reflecting and translating a triangle

Problem

Triangle ABC has vertices A(1,2), B(4,2), C(4,5). (a) Reflect across the y-axis. (b) Then translate by (−3, −1).

Worked Example

Reflecting across the lines y = x and y = −x

Problem

Point P(3,1)P(3, 1). Reflect across (a) y=xy = x and (b) y=−xy = -x.

Worked Example

Translation using vector notation

Problem

Quadrilateral ABCD: A(1,4), B(3,4), C(4,2), D(2,2). Translate by vector (−4−3)\begin{pmatrix}-4\\-3\end{pmatrix}. Find new vertices and state which quadrant each original vertex is in.
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge2 marks
Reflect the point (3,−5)(3, −5) across the x-axis.
2
L1 · Knowledge2 marks
Translate the point (2,7)(2, 7) by the vector (5,−3)(5, −3).
3
L2 · Routine Procedures3 marks
Point P(4, 1) is reflected across the line y = x. Find P'.
4
L2 · Routine Procedures4 marks
Triangle DEF has D(−2, 3), E(1, 3), F(−2, 0). Translate by (4, −2). State the new coordinates.
5
L3 · Complex Procedures5 marks
Point A(3, 4) is reflected first across the y-axis, then across the x-axis. Find the final image and describe the single transformation equivalent.
6
L3 · Complex Procedures4 marks
A point P and its reflection P' across the line y = x are equidistant from the line. If P = (a, b), show that the midpoint of PP' lies on y = x.
7
L4 · Problem Solving5 marks
Square ABCD has A(1,1), B(3,1), C(3,3), D(1,3). (a) Reflect across y=−x. (b) What symmetry does the image have with the original?
8
L4 · Problem Solving4 marks
A translation maps A(2, 5) to A'(−1, 3). What vector describes this translation? Apply the same translation to B(−3, 0) and find B'.
Weeks 8–9

11.2 Rotations and Enlargements/Reductions

  • Rotate figures by 90°, 180°, and 270° about the origin
  • Enlarge and reduce figures from the origin or a given centre using a scale factor
  • Determine the scale factor from an original and its image
  • Identify the properties preserved under each transformation
🌍

Real-World Connection

Rotations appear in wheels turning, clock hands moving, and Earth spinning on its axis. Enlargements appear in projectors (enlarging images), microscopes (enlarging specimens), and photocopiers (scaling documents). In computer graphics, every time you zoom in on a map, an enlargement is applied about the centre of your screen.

Rotation Rules (about Origin)

Rotation

Coordinate Rule

90° anticlockwise

(x, y) → (−y, x)

90° clockwise

(x, y) → (y, −x)

180°

(x, y) → (−x, −y)

Same as 180° either direction

270° anticlockwise

(x, y) → (y, −x)

Same as 90° clockwise

Enlargement / Reduction

(x,y)→(kx,ky)(x, y) \to (kx, ky)

k = scale factor. k > 1: enlargement. 0 < k < 1: reduction. Centre of enlargement is the origin.

Property / Rule

Effect of Enlargement/Reduction on Side Lengths

When a figure is enlarged or reduced by scale factor k, every side length of the image is k times the corresponding side of the original. The shapes are SIMILAR (equal angles, proportional sides).

A′B′=k⋅AB,B′C′=k⋅BC,A′C′=k⋅ACA'B' = k \cdot AB, \quad B'C' = k \cdot BC, \quad A'C' = k \cdot AC

Property / Rule

Effect on Perimeter

Perimeter is a linear measure (sum of side lengths). Since every side scales by k, the perimeter also scales by k.

Perimeter of image=k×Perimeter of original\text{Perimeter of image} = k \times \text{Perimeter of original}

Property / Rule

Effect on Area

Area is a two-dimensional measure. Since BOTH dimensions scale by k, area scales by k². This is true for ALL shapes — triangles, rectangles, circles.

Area of image=k2×Area of original\text{Area of image} = k^2 \times \text{Area of original}

Summary: Properties Preserved Under Each Transformation

Transformation

Size | Shape | Orientation | Centre preserved

Translation

Yes — congruent (isometry)

✓ size ✓ shape ✓ orientation — all preserved

Reflection

Yes — congruent (isometry)

✓ size ✓ shape ✗ orientation (mirror reverses)

Rotation

Yes — congruent (isometry)

✓ size ✓ shape ✓ orientation — all preserved

Enlargement (k≠1)

No — similar only

✗ size ✓ shape ✓ angles — perimeter ×k, area ×k²

ℹ️ Note

Enlargements change size but PRESERVE shape and angles (similar figures). Rotations, reflections, and translations preserve BOTH size and shape (congruent figures, called isometries). Enlargements are the ONLY transformation that is NOT an isometry.

⚠️ Warning

When k = 1 (scale factor of 1), an enlargement leaves the figure unchanged — it is effectively an identity transformation. When 0 < k < 1, the image is smaller (a REDUCTION). When k > 1, the image is larger (an ENLARGEMENT).

Worked Examples

Worked Example

Area and perimeter under enlargement

Problem

△ABC has sides AB=4 cm, BC=3 cm, AC=5 cm, and area =6 cm2.It is enlarged from the origin by k=3. Find: (a) side lengths of image, (b) perimeter of image, (c) area of image.\triangle ABC \text{ has sides } AB=4\text{ cm},\ BC=3\text{ cm},\ AC=5\text{ cm, and area }=6\text{ cm}^2. \\[4pt] \text{It is enlarged from the origin by } k=3. \text{ Find: (a) side lengths of image, (b) perimeter of image, (c) area of image.}

Worked Example

Rotating and enlarging

Problem

Triangle PQR has P(2,1), Q(5,1), R(5,4). (a) Rotate 90° anticlockwise. (b) Enlarge by scale factor 2 from the origin.

Worked Example

All rotation rules and 270°

Problem

Point A(4,−2)A(4, -2). Apply: (a) 90° clockwise, (b) 180°, (c) 270° clockwise about the origin.

Worked Example

Reduction (scale factor 0 < k < 1)

Problem

Rectangle PQRS: P(0,0), Q(6,0), R(6,4), S(0,4). Reduce by scale factor k=12k = \frac{1}{2} from the origin. Find the new vertices and compare areas.
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge2 marks
Rotate the point (3,5)(3, 5) by 180°180° about the origin.
2
L1 · Knowledge2 marks
A shape is enlarged from the origin by scale factor 3. Point A was at (2,4)(2, 4). Where is A'?
3
L2 · Routine Procedures3 marks
Rotate point Q(4, −2) by 90° clockwise about the origin.
4
L2 · Routine Procedures3 marks
The original point A(6, 8) maps to A'(3, 4) under an enlargement from the origin. Find the scale factor.
5
L3 · Complex Procedures5 marks
Triangle XYZ has X(1,0), Y(3,0), Z(2,2). Rotate 270° anticlockwise. Find X', Y', Z' and describe the position of the new triangle.
6
L3 · Complex Procedures5 marks
Shape ABCD has A(2,1), B(5,1), C(5,4), D(2,4). Enlarge by factor 2 from centre (1,1) (not the origin).
7
L4 · Problem Solving5 marks
A figure is first rotated 90° anticlockwise, then enlarged by scale factor 2. If the original point is (3, 1), find the final image. Is the order of transformations important?
8
L4 · Problem Solving4 marks
Rectangle PQRS has area 20 cm². After an enlargement by scale factor 3, what is the area of P'Q'R'S'?

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Transformation Geometry Grade 9 Maths CAPS Notes & Examples | MathSciBuddy