5.2.8 · Higher Tier
Combined Transformations
The Sequence — Master the Order Rule & Shortcut Theorems
🎯 EDEXCEL 4MA1
Shape, Space & Measures
⚙ Sequence Simulator
Show steps:
⚡
HUD ALERT: Shortcut Detected!
Wait! Two parallel reflections is just one big Translation!
Original
After Reflect x=0
After Reflect x=3
x = 0
x = 3
● Phase 2: The Twist
Double Reflection = Translation
Reflecting a shape in two parallel mirror lines is equivalent to a single translation. The translation vector is twice the perpendicular distance between the mirrors.
📐 TRANSFORMATION RULES
Translation = 2 × (distance between mirrors)
⚡ SHORTCUT THEOREM
Two parallel reflections → One Translation
⚠ EXAMINER TRAP
Students often think two reflections create another reflection. WRONG! Two parallel reflections always combine into a translation.
📋 THE ORDER RULE
Rule 1: Order Matters!
Transformation A then B means do A FIRST, then apply B to the new image.
Rule 2: Parallel Lines → Translation
Reflect across two parallel lines? That is a translation equal to twice the distance between them.
Rule 3: Intersecting Lines → Rotation
Reflect across two intersecting lines? That is a rotation by twice the angle between them.
Dynamic Solver
Sequence Analyser — Double Reflection
Step 1
Identify the two mirror lines
Line 1: x = 0 | Line 2: x = 3
Step 2
Find perpendicular distance between mirrors
Distance = 3 − 0 = 3 units
Step 3
Calculate the combined translation
Translation = 2 × 3 = 6 units in +x direction
Result
Single equivalent transformation
Translate by vector (6, 0)
📐 PROOF — PARALLEL REFLECTIONS → TRANSLATION
Point P(x, y). Mirror 1: x = 0. Mirror 2: x = 3.
After reflect x = 0 : P → (−x, y)
After reflect x = 3 : (−x, y) → (6 − (−x), y) = (6+x, y)
Net change: x → x + 6 ∴ Translation by (6, 0) ✓
🟢 Phase 1 — Anchor (Grade 4-5)
Two translations combine into one. Add the vectors. Simple, always works.
🟡 Phase 2 — Twist (Grade 6-7)
Two parallel reflections give a translation. The distance doubles. Many students miss this shortcut.
🔴 Phase 3 — Killer (Grade 8-9)
Mixed sequences (reflect + rotate). Prove the combined transformation. Order of operations must be tracked precisely.
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