Testing and justifying solutions | Year 10

01 / LEARN

Understand the idea.

A test can disprove a claim

To test an “always” claim, separate its conditions from its conclusion. A counterexample must satisfy every condition and still make the conclusion false; one such example disproves the universal statement.

Check that your example is allowed. Triangle angles must be positive and total $180^{\circ}$ ; points required to lie on a circle must actually do so. An excluded degenerate case cannot refute the claim.

Validate the chain and the answer

Validate a solution on two levels: check the arithmetic and check why each theorem applies. Match corresponding vertices, select the correct intercepted arc, and distinguish a radius from half a chord.

Use substitution or an independent relationship to test a proposed number. For constructions, vary size, orientation and boundary inputs. These checks can expose an error, while a general justification explains why the method works throughout its domain.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Disprove a similarity claim

A student claims any two triangles containing a 60° angle are similar. Give a counterexample.

Worked solution

1
Choose two valid angle sets.
$\begin{aligned} 60^{\circ} + 60^{\circ} + 60^{\circ} & = 180^{\circ} \\ 60^{\circ} + 50^{\circ} + 70^{\circ} & = 180^{\circ} \end{aligned}$
Each set contains positive angles summing to 180 degrees, so each describes a possible triangle.
2
Check the stated condition.
Both triangles contain a 60-degree angle, as the claim requires.
3
Show the conclusion fails.
${60^{\circ}, 60^{\circ}, 60^{\circ}} \neq {60^{\circ}, 50^{\circ}, 70^{\circ}}$
Similar triangles have the same angle set. These triangles satisfy the condition but are not similar.

Answer

The two listed triangles disprove the claim.

EXAMPLE 02

Check a chord result

A radius 5 cm circle has an 8 cm chord. A student reports a centre-to-chord distance 4 cm. Check the answer.

Worked solution

1
Identify the legs and hypotenuse.
$\frac{8}{2} = 4 cm, r = 5 cm$
The perpendicular from the centre bisects the chord, making a right triangle with half-chord 4 cm and hypotenuse 5 cm.
2
Test the proposed distance.
$4^{2} + 4^{2} = 32 \neq 25 = 5^{2}$
A distance of 4 cm does not satisfy Pythagoras with the given radius.
3
Calculate the correct distance.
$d^{2} = 5^{2} - 4^{2} = 9$
Subtract the known leg’s square from the hypotenuse’s square.
4
Check the positive result.
$d = 3 cm, 3^{2} + 4^{2} = 5^{2}$
A distance is positive, and substitution confirms the original right-triangle relationship.

Answer

The answer is 3 cm, not 4 cm.

03 / YOUR TURN

Try it yourself.

Check your understanding

Does one valid counterexample disprove an “always” statement?

Show answer & reasoning

Yes.

Yes. An “always” statement includes every allowed case, so one allowed case with a false conclusion is sufficient to disprove it.

Keep practising

6 worksheet questions · 15 marks

Open worksheet Worked solutions

Read the questions online

Disprove: all quadrilaterals with four equal sides have four right angles.
3 marks
A claimed triangle has sides 3, 4, 8. Explain why it cannot serve as a triangle example.
2 marks
A cyclic quadrilateral has opposite angles $85^{\circ}$ and $95^{\circ}$ . Is this information consistent?
2 marks
A radius 10 circle has a chord 12 long. A student finds its centre distance 8. Check it.
2 marks
A square-construction algorithm is tested only on horizontal sides. Give another important test and explain.
3 marks
Why must corresponding vertices be named consistently when using triangle similarity?
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Disprove: all quadrilaterals with four equal sides have four right angles.
3 marks
Its sides are equal but its angles need not be $90^{\circ}$ .
A non-square rhombus.
A claimed triangle has sides 3, 4, 8. Explain why it cannot serve as a triangle example.
2 marks
$3 + 4 = 7 < 8$ .
It violates the triangle inequality.
A cyclic quadrilateral has opposite angles $85^{\circ}$ and $95^{\circ}$ . Is this information consistent?
2 marks
$85 + 95 = 180^{\circ}$ .
Yes.
A radius 10 circle has a chord 12 long. A student finds its centre distance 8. Check it.
2 marks
Half-chord 6.
$6^{2} + 8^{2} = 100 = 10^{2}$ .
Correct.
A square-construction algorithm is tested only on horizontal sides. Give another important test and explain.
3 marks
A correct geometric construction should not depend on page orientation.
This can reveal hidden horizontal/vertical assumptions.
Use a sloping side.
Why must corresponding vertices be named consistently when using triangle similarity?
3 marks
A wrong ordering can produce ratios between non-corresponding sides and an invalid answer.
They determine which angles and sides match.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum

Angle and similarity reasoning; congruent/similar triangles in proofs; chords, radii and tangents; angles at the centre and circumference; circle theorems and their applications; connected deductive arguments; construction/spatial algorithms; testing and justifying solutions.

Official curriculum source ↗

In this lesson

Understand the idea.

A test can disprove a claim

Validate the chain and the answer

See the method in action.

Disprove a similarity claim

Choose two valid angle sets.

Check the stated condition.

Show the conclusion fails.

Check a chord result

Identify the legs and hypotenuse.

Test the proposed distance.

Calculate the correct distance.

Check the positive result.

Try it yourself.

Check your understanding

Keep practising