The idea, step by step
Test different inputs
A procedure may work accidentally on a special input. Try horizontal, vertical and sloping segments; positive and negative coordinates; different lengths; and scale factors both above and below 1. State an expected result before testing.
A bisector must both halve the segment and meet it at 90°. An enlargement must have the right distance ratio and keep original/image points aligned with the centre. Looking similar is not a sufficient check.
Locate and repair the fault
Record input, actual output and the failed property. Find the first step that causes the problem, change it, then repeat both the previously failed case and earlier successful cases.
A finite list of successes is not a general proof. Use equal-radius reasoning or algebra to explain why the corrected procedure works for every allowed input.
A common trap
One successful test proves a procedure.
Instead: A special input may hide an error. Test varied inputs, then explain why the corrected steps work generally.
See the thinking
A boundary case
A bisector procedure allows r=L/2. Test a 6 cm segment.
- The allowed radius is 3 cm.
- The circles touch once, not twice, so no joining line is determined.
- Require r>L/2 instead. Radius 4 cm gives two crossings.
Repair a midpoint formula
Test the proposed midpoint ((x₂−x₁)/2,(y₂−y₁)/2) on A(2,1), B(8,5).
- The proposed point is (3,2).
- The true midpoint is ((2+8)/2,(1+5)/2)=(5,3).
- The wrong rule finds half a displacement. Add A back, or average the coordinates.
Check your understanding
A bisector procedure uses radius 2 cm for a segment of 6 cm. What needs fixing?
Show answer
Choose radius greater than 3 cm.
The circles need to intersect twice.
Your turn
6 questions · 18 marks · Work at your own pace.
Read the questions online
- Test radius r=5 cm on segment AB=10 cm for the usual two-crossing bisector.3 marks
- A factor-2 enlargement rule doubles x but leaves y unchanged. Test P(2,3).3 marks
- Why is P(4,0) a poor only test for the faulty factor-2 rule (x,y)→(2x,y)?3 marks
- Test the midpoint rule ((x₁+x₂)/2,(y₁+y₂)/2) on A(−4,2), B(2,8).2 marks
- A factor-1/2 rule rounds every image coordinate to a whole number. Test P(1,3) and refine the rule.3 marks
- Give three useful types of test for a bisector construction and explain why tests alone are not a proof.4 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Test radius r=5 cm on segment AB=10 cm for the usual two-crossing bisector.3 marks
The chosen radius equals half the segment.
Two distinct crossings require more than half.
It fails: the circles touch once. Choose r>5 cm. - A factor-2 enlargement rule doubles x but leaves y unchanged. Test P(2,3).3 marks
An enlargement from the origin scales both coordinates.
Changing only one coordinate distorts the figure.
It gives (4,3), but the correct image is (4,6). - Why is P(4,0) a poor only test for the faulty factor-2 rule (x,y)→(2x,y)?3 marks
Doubling zero leaves it zero.
Use a point with nonzero y, such as (1,2).
Both the faulty and correct rules give (8,0). - Test the midpoint rule ((x₁+x₂)/2,(y₁+y₂)/2) on A(−4,2), B(2,8).2 marks
Average x to get −1.
Average y to get 5.
M(−1,5). - A factor-1/2 rule rounds every image coordinate to a whole number. Test P(1,3) and refine the rule.3 marks
Whole-number rounding changes the required distance ratio and direction.
Keep exact halves or use a half-unit grid.
The exact image is (0.5,1.5); retain these values. - Give three useful types of test for a bisector construction and explain why tests alone are not a proof.4 marks
Choose inputs that expose different possible faults.
A general argument must explain why equal-radius crossings give the required line.
Vary segment length, direction and radii near the half-length boundary. Tests check selected cases only.
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 ↗