Testing/refining construction algorithms | Year 9

01 / LEARN

Understand the idea.

Test different inputs

Test an algorithm against its required properties, rather than whether the drawing looks plausible. A perpendicular bisector must meet the segment at $90^{\circ}$ and divide it equally. An enlargement must keep each point aligned with the centre and scale its distance correctly.

Choose varied inputs that might expose a hidden assumption: horizontal, vertical and sloping segments; different lengths; negative coordinates; and enlargement factors above and below $1$ . Include boundary cases, such as a radius equal to half a segment.

Locate and repair the fault

When a test fails, record the input, the actual output and the first property that fails. Find the step responsible, repair it, then repeat the failed case and earlier successful cases to check that the change has not introduced another fault.

One counterexample disproves a universal claim, but a finite set of successful tests does not prove it. Explain the corrected procedure generally using equal-distance geometry or algebra, including the conditions under which that reasoning applies.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A boundary case

A bisector procedure allows r=L/2. Test a 6 cm segment.

Worked solution

1
Test the allowed boundary radius.
$L = 6 cm, r = \frac{L}{2} = 3 cm$
The proposed rule permits a radius exactly half the segment length.
2
Check the number of intersections.
$r + r = 3 + 3 = 6 cm = L$
The equal circles just touch at the midpoint, giving one crossing rather than two.
3
Explain the failure.
One point does not determine a unique line. There are no two distinct crossings to join, so the construction cannot produce its required bisector.
4
Strengthen and test the condition.
$r > \frac{L}{2}; r = 4 cm > 3 cm$
For distinct endpoints, equal circles with radius greater than half their separation intersect twice; $4 cm$ satisfies this condition.

Answer

Replace “at least half” with “greater than half”.

EXAMPLE 02

Repair a midpoint formula

Test the proposed midpoint ((x₂−x₁)/2,(y₂−y₁)/2) on A(2,1), B(8,5).

Worked solution

1
Evaluate the proposed formula.
$(\frac{8 - 2}{2}, \frac{5 - 1}{2}) = (3, 2)$
This calculates half the displacement from $A$ to $B$ , rather than its position from the origin.
2
Add the starting point back.
$M = (2, 1) + (3, 2) = (5, 3)$
Moving halfway from $A$ requires starting at $A$ before adding the half-displacement.
3
Write the corrected general rule.
$M = A + \frac{B - A}{2} = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$
Combining $A$ with half of $B - A$ averages the endpoint coordinates for any two points.
4
Check both displacement halves.
$\begin{aligned} M - A & = (5 - 2, 3 - 1) = (3, 2) \\ B - M & = (8 - 5, 5 - 3) = (3, 2) \end{aligned}$
The equal successive displacements show that $(5, 3)$ lies halfway along this segment.

Answer

Use (5,3); the correct rule is A+(B−A)/2.

03 / YOUR TURN

Try it yourself.

Check your understanding

A bisector procedure uses radius 2 cm for a segment of 6 cm. What needs fixing?

Show answer & reasoning

Choose radius greater than 3 cm.

For a $6 cm$ segment, half the length is $3 cm$ . Equal radii of $2 cm$ cannot meet; the procedure needs a shared radius greater than $3 cm$ .

Keep practising

6 worksheet questions · 18 marks

Open worksheet Worked solutions

Read the questions online

Test radius $r = 5$ cm on segment $A B = 10$ 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 $(x, y) \mapsto (2 x, y)$ ?
3 marks
Test the midpoint rule $(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$ on $A (- 4, 2)$ , $B (2, 8)$ .
2 marks
A factor- $\frac{1}{2}$ 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 $A B = 10$ 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 $(x, y) \mapsto (2 x, y)$ ?
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 $(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$ on $A (- 4, 2)$ , $B (2, 8)$ .
2 marks
Average $x$ to get $- 1$ .
Average $y$ to get 5.
$M (- 1, 5)$ .
A factor- $\frac{1}{2}$ 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.

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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 ↗

In this lesson

Understand the idea.

Test different inputs

Locate and repair the fault

See the method in action.

A boundary case

Test the allowed boundary radius.

Check the number of intersections.

Explain the failure.

Strengthen and test the condition.

Repair a midpoint formula

Evaluate the proposed formula.

Add the starting point back.

Write the corrected general rule.

Check both displacement halves.

Try it yourself.

Check your understanding

Keep practising