The idea, step by step
Make each step testable
A geometric algorithm is a finite sequence that produces a specified result from stated input. Name starting objects, tools and order. Replace “draw it neatly” with an instruction such as “join the two arc crossings”.
A decision changes the next action. In a bisector algorithm, check whether the radius is greater than half the segment length; if not, increase it before drawing arcs.
Order and repetition matter
Translating then enlarging can differ from enlarging then translating. Trace one point through all steps before applying the procedure to a complete figure.
For repetition, state what repeats, how many times and when to stop. A regular hexagon can be made by stepping one radius around a circle six times, then joining neighbouring marked points.
A common trap
A list of transformations works in any order.
Instead: Order is part of the rule; test a point instead of assuming the steps commute.
See the thinking
Trace an ordered pair of moves
Start at P(1,2), translate by (2,1), then enlarge by factor 2 from the origin.
- Translation gives (3,3).
- Enlargement gives (6,6).
- Reversing the order gives (2,4)+(2,1)=(4,5), so order matters.
A construction decision
Give the radius decision for a bisector of a segment of length L.
- Read L and chosen radius r.
- If r≤L/2, choose a larger radius and check again.
- When r>L/2, draw equal arcs from both endpoints, join their two crossings and stop.
Check your understanding
Translate (2,−1) by (1,3), then double its coordinates. Find the output.
Show answer
(6,4)
The intermediate point is (3,2).
Your turn
6 questions · 15 marks · Work at your own pace.
Read the questions online
- Start at (−1,2), translate by (3,−1), then enlarge by factor 2 from the origin.2 marks
- From (0,0), repeat “move 2 units right” four times. Find the final point.2 marks
- Complete the decision: for segment length L, increase radius r if ___.2 marks
- Enlarge P(1,1) by factor 3 from the origin, then translate by (2,0). Find the image.2 marks
- Describe a finite compass algorithm to mark a regular hexagon on a circle.4 marks
- An algorithm says “if four sides are equal, the quadrilateral is a square”. Give a counterexample and repair the decision.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Start at (−1,2), translate by (3,−1), then enlarge by factor 2 from the origin.2 marks
Translation gives (2,1).
Enlargement gives (4,2).
(4,2) - From (0,0), repeat “move 2 units right” four times. Find the final point.2 marks
Four moves give horizontal change 4×2=8, with no vertical change.
(8,0) - Complete the decision: for segment length L, increase radius r if ___.2 marks
The radius must be greater than half the segment to produce two arc crossings.
r≤L/2 - Enlarge P(1,1) by factor 3 from the origin, then translate by (2,0). Find the image.2 marks
Enlargement gives (3,3).
Translation gives (5,3).
(5,3) - Describe a finite compass algorithm to mark a regular hexagon on a circle.4 marks
Keep the compass width equal to the radius.
Six equal chords close the hexagon.
Stop after six sides.
Choose a starting point, step the circle radius around the circumference six times, then join neighbouring vertices. - An algorithm says “if four sides are equal, the quadrilateral is a square”. Give a counterexample and repair the decision.3 marks
Equal sides alone do not force right angles.
A quadrilateral with all sides equal and a right angle is a square.
A non-square rhombus; also require a right angle.
Curriculum reference & lesson scope
Australian Curriculum v9 / South Australian Curriculum
Angle reasoning consolidation; enlargements using coordinates/digital geometry; invariant and changing properties; geometric constructions; sequences of steps and decisions; testing/refining construction algorithms; explaining geometric reasoning.
Official curriculum source ↗