Transformation Geometry — Grade 9 Maths (CAPS)

Reflecting, translating, rotating, enlarging and reducing whole figures on the Cartesian plane, the coordinate rules for each, working backwards from an image, and the effect of enlargement on perimeter and area.

Also known as: translation, reflection, rotation · enlargement · dilation

Grade 8 transformed single points. Grade 9 transforms whole figures — and the trick is simple: transform each vertex with the coordinate rule, then join them up again. The shape looks after itself.

1The coordinate rules

Reflect in the x-axis: (x ; y) → (x ; −y)
Reflect in the y-axis: (x ; y) → (−x ; y)
Reflect in the line y = x: (x ; y) → (y ; x)
Rotate 90° clockwise about the origin: (x ; y) → (y ; −x)
Rotate 90° anticlockwise about the origin: (x ; y) → (−y ; x)
Rotate 180° about the origin: (x ; y) → (−x ; −y)
A turn of 270° is the same quarter turn the other way: 270° clockwise = 90° anticlockwise, and 270° anticlockwise = 90° clockwise.

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Remember which letter changes: reflecting in the x-axis changes the y-value, and reflecting in the y-axis changes the x-value. It feels backwards, which is exactly why it is examined.

2Reflecting a whole figure

Every vertex is reflected, then the image is joined up. The image is congruent to the original — same size, same shape.

Worked ExampleReflect △A with vertices (1 ; 1), (4 ; 1), (1 ; 4) in the y-axis

1
Apply (x ; y) → (−x ; y) to each vertex.
Reflecting in the y-axis flips the sign of x.
2
(1 ; 1) → (−1 ; 1), (4 ; 1) → (−4 ; 1), (1 ; 4) → (−1 ; 4).
3
Join the three image points to draw △A′.
The image is congruent to the original.

3Translating a figure

A translation slides every point the same distance in the same direction.

Worked ExampleTranslate (5 ; 3) by 2 left and 4 up

1
Left means subtract from x: 5 − 2 = 3.
Left/down subtract; right/up add.
2
Up means add to y: 3 + 4 = 7.
3
Image: (3 ; 7).

4Rotating about the origin — and about another point

A rotation turns the figure about a point. Most questions turn about the origin, but the ATP also asks you to rotate about a given point — the last example in this section shows how. The turns are multiples of a quarter turn: 90°, 180° and 270°. Rather than memorising four separate rules, look at what one quarter turn does on the grid:

One point, three rotations about the origin. A quarter turn swaps the two coordinates and changes ONE sign; a half turn changes both signs and swaps nothing.

Worked ExampleRotate P(3 ; 2) 90° CLOCKWISE about the origin

1
Swap the coordinates: (3 ; 2) becomes (2 ; 3).
A quarter turn always swaps them — what was across becomes up, and what was up becomes across.
2
Clockwise, so make the new y-coordinate negative: (2 ; −3).
The rule is (x ; y) → (y ; −x).
3
Image: (2 ; −3).
Check on the sketch: P is up and to the right; a quarter turn clockwise lands it down and to the right.

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Which sign changes? Turn the page and watch the quadrant. Clockwise from the first quadrant lands you in the fourth (x positive, y negative), so the minus belongs on the second coordinate. Anticlockwise lands you in the second quadrant, so the minus belongs on the first. Checking the quadrant beats memorising the rule.

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Rotating 90° without swapping the coordinates. Only the 180° rule keeps x and y in place — every quarter turn swaps them first, and THEN changes one sign.

Worked ExampleRotate P(4 ; 3) 90° CLOCKWISE about C(1 ; 1)

1
Measure P from C: P is 3 right and 2 up from C, so use (3 ; 2).
Subtract C from P: (4 − 1 ; 3 − 1). C now plays the part of the origin.
2
Rotate (3 ; 2) by a quarter turn clockwise: (2 ; −3).
The usual rule (x ; y) → (y ; −x).
3
Add C back: (2 + 1 ; −3 + 1) = (3 ; −2).
Check: CP and CP′ are both √13 long, and they are at right angles.

5Working backwards from the image

Worked ExampleA point is reflected in the x-axis and its image is (6 ; −2). Find the original

1
The rule is (x ; y) → (x ; −y).
2
Reversing it changes the sign of y again.
Reflections undo themselves.
3
The original point was (6 ; 2).
Check: reflecting (6 ; 2) in the x-axis gives (6 ; −2) ✓

6Enlargements and reductions

An enlargement from the origin with scale factor k multiplies every coordinate by k. Unlike the other transformations, this one changes the size, so the image is similar but not congruent.

Enlargement by factor 2 from the origin: every coordinate is doubled.

Enlargement by factor k: (x ; y) → (kx ; ky)
Perimeter × k · Area × k²

Worked ExampleA rectangle 4 cm by 3 cm is enlarged by scale factor 3. Find the new perimeter and area

1
New sides: 4 × 3 = 12 cm and 3 × 3 = 9 cm.
Lengths multiply by k.
2
New perimeter = 2(12 + 9) = 42 cm.
Original was 2(4 + 3) = 14, and 14 × 3 = 42 ✓
3
New area = 12 × 9 = 108 cm².
Original was 12 cm², and 12 × 3² = 108 ✓

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Area does not scale by k — it scales by k². Enlarging by 3 makes the area 9 times as big, not 3 times.

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Reflections, translations and rotations produce congruent images (size unchanged). Only enlargements and reductions change size, giving a similar image.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Reflect (3 ; 5) in the x-axis.
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(3 ; −5)
2
Reflect (−2 ; 7) in the y-axis.
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(2 ; 7)
3
Reflect (4 ; 1) in the line y = x.
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(1 ; 4)
4
Rotate (3 ; −6) by 180° about the origin.
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(−3 ; 6)
5
Rotate (4 ; 1) by 90° CLOCKWISE about the origin.
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Swap, then make the second coordinate negative: (x ; y) → (y ; −x) (1)
(1 ; −4) (1)
6
Rotate (−2 ; 5) by 90° ANTICLOCKWISE about the origin.
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(x ; y) → (−y ; x) (1)
(−5 ; −2) (1)
7
Rotate (3 ; −1) by 270° clockwise about the origin.
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270° clockwise is the same as 90° anticlockwise (1)
(x ; y) → (−y ; x), so the image is (1 ; 3) (1)
8
Translate (5 ; 3) by 2 left and 4 up.
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(3 ; 7)
9
A point is reflected in the x-axis; its image is (6 ; −2). Find the original.
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(6 ; 2)
10
Enlarge (3 ; −4) by scale factor 3 from the origin.
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(9 ; −12)
11
A rectangle 4 cm × 3 cm is enlarged by factor 3. Find the new area.
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12 × 3² = 108 cm²
12
Which transformations leave the figure congruent?
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Reflection, translation and rotation (enlargement changes size).