Transformation Geometry (Grade 9)

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.

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 four 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 180° about the origin: (x ; y) → (−x ; −y)

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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).

4Working 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) ✓

5Enlargements 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 bigger, 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
Translate (5 ; 3) by 2 left and 4 up.
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(3 ; 7)
6
A point is reflected in the x-axis; its image is (6 ; −2). Find the original.
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(6 ; 2)
7
Enlarge (3 ; −4) by scale factor 3 from the origin.
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(9 ; −12)
8
A rectangle 4 cm × 3 cm is enlarged by factor 3. Find the new area.
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12 × 3² = 108 cm²
9
Which transformations leave the figure congruent?
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Reflection, translation and rotation (enlargement changes size).