Australian Curriculum v9 / ACiQ Year 8 Mathematics - Unit 3 - Pythagoras' theorem

Pythagoras' theorem

Use Pythagoras' theorem to find unknown side lengths and test whether triangles are right-angled.

Updated 2026-07-26 - 12 min read

Pythagoras' theorem is taught here as a connected set of decisions, not a list of facts. Work through the prerequisite recall, explicit models, carefully faded examples, misconception repairs and transfer task before using the target in Check, Practice, Review or Rapid Revision.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Identify the hypotenuse and legs

In a right triangle, the hypotenuse is opposite the right angle and is always the longest side; the other sides are legs. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by identify the hypotenuse and legs and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Position and drawing scale are unreliable; identify the side opposite the marked 90° angle.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

In a right triangle, side c is opposite the 90° angle. Which side is the hypotenuse?

Step 1 - identify the governing idea: In a right triangle, the hypotenuse is opposite the right angle and is always the longest side; the other sides are legs.

Step 2 - apply it to this evidence: The hypotenuse is defined by its position opposite the right angle.

Result: c

The relationship is visible in the working: The hypotenuse is defined by its position opposite the right angle. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • The shortest side — It does not agree with the required relationship: The hypotenuse is defined by its position opposite the right angle.
  • Whichever side is vertical — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • It cannot be identified — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

A right triangle has sides 6 cm, 8 cm and 10 cm. Which are the legs?

Step 1 - identify the governing idea: In a right triangle, the hypotenuse is opposite the right angle and is always the longest side; the other sides are legs.

Step 2 - apply it to this evidence: The longest side 10 cm is the hypotenuse.

Result: 6 cm and 8 cm

The relationship is visible in the working: The longest side 10 cm is the hypotenuse. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 6 cm and 10 cm — It does not agree with the required relationship: The longest side 10 cm is the hypotenuse.
  • 8 cm and 10 cm — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • All three sides — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Which equation matches legs a and b and hypotenuse c?

Step 1 - identify the governing idea: In a right triangle, the hypotenuse is opposite the right angle and is always the longest side; the other sides are legs.

Step 2 - apply it to this evidence: Pythagoras relates the sum of squared legs to the squared hypotenuse.

Result: a² + b² = c²

The relationship is visible in the working: Pythagoras relates the sum of squared legs to the squared hypotenuse. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • a + b = c — It does not agree with the required relationship: Pythagoras relates the sum of squared legs to the squared hypotenuse.
  • a² − b² = c² — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2a + 2b = c — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Find unknown right-triangle sides

Substitute known lengths into a² + b² = c², isolate the unknown square and take the positive square root. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by find unknown right-triangle sides and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: The theorem adds squares of the legs; when a leg is unknown, subtract its known squared leg from the hypotenuse squared.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

Find the hypotenuse when the legs are 9 cm and 12 cm.

Step 1 - identify the governing idea: Substitute known lengths into a² + b² = c², isolate the unknown square and take the positive square root.

Step 2 - apply it to this evidence: c = √(9² + 12²) = √225 = 15.

Result: 15 cm

The relationship is visible in the working: c = √(9² + 12²) = √225 = 15. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 21 cm — It does not agree with the required relationship: c = √(9² + 12²) = √225 = 15.
  • 10.5 cm — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 225 cm — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

A hypotenuse is 13 m and one leg is 5 m. Find the other leg.

Step 1 - identify the governing idea: Substitute known lengths into a² + b² = c², isolate the unknown square and take the positive square root.

Step 2 - apply it to this evidence: The leg is √(13² − 5²) = √144 = 12.

Result: 12 m

The relationship is visible in the working: The leg is √(13² − 5²) = √144 = 12. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 18 m — It does not agree with the required relationship: The leg is √(13² − 5²) = √144 = 12.
  • √194 m — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 8 m — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

A 4 m ladder reaches 3.2 m up a wall. How far is its foot from the wall?

Step 1 - identify the governing idea: Substitute known lengths into a² + b² = c², isolate the unknown square and take the positive square root.

Step 2 - apply it to this evidence: The ladder is the hypotenuse, so distance = √(4² − 3.2²) = √5.76 = 2.4.

Result: 2.4 m

The relationship is visible in the working: The ladder is the hypotenuse, so distance = √(4² − 3.2²) = √5.76 = 2.4. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 7.2 m — It does not agree with the required relationship: The ladder is the hypotenuse, so distance = √(4² − 3.2²) = √5.76 = 2.4.
  • 0.8 m — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 5.12 m — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Test whether a triangle is right-angled

Order the sides so c is longest, then compare a² + b² with c²; equality means the triangle is right-angled. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by test whether a triangle is right-angled and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Pythagoras is a test specifically for right triangles; equality must be checked using the longest side as c.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

Are sides 7, 24 and 25 a right triangle?

Step 1 - identify the governing idea: Order the sides so c is longest, then compare a² + b² with c²; equality means the triangle is right-angled.

Step 2 - apply it to this evidence: 7² + 24² = 49 + 576 = 625 = 25².

Result: Yes

The relationship is visible in the working: 7² + 24² = 49 + 576 = 625 = 25². Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • No — It does not agree with the required relationship: 7² + 24² = 49 + 576 = 625 = 25².
  • Only if 7 is the hypotenuse — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • There is not enough information — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Are sides 6, 8 and 11 a right triangle?

Step 1 - identify the governing idea: Order the sides so c is longest, then compare a² + b² with c²; equality means the triangle is right-angled.

Step 2 - apply it to this evidence: 6² + 8² = 100, which is not 11² = 121.

Result: No

The relationship is visible in the working: 6² + 8² = 100, which is not 11² = 121. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Yes — It does not agree with the required relationship: 6² + 8² = 100, which is not 11² = 121.
  • Only after rounding — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Yes, because 6 + 8 > 11 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

A rectangle is 8 m by 15 m. Find its diagonal.

Step 1 - identify the governing idea: Order the sides so c is longest, then compare a² + b² with c²; equality means the triangle is right-angled.

Step 2 - apply it to this evidence: The diagonal is the hypotenuse: √(8² + 15²) = √289 = 17.

Result: 17 m

The relationship is visible in the working: The diagonal is the hypotenuse: √(8² + 15²) = √289 = 17. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 23 m — It does not agree with the required relationship: The diagonal is the hypotenuse: √(8² + 15²) = √289 = 17.
  • 7 m — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 120 m — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. A right triangle has sides 6 cm, 8 cm and 10 cm. Which are the legs?
  2. A hypotenuse is 13 m and one leg is 5 m. Find the other leg.
  3. Are sides 6, 8 and 11 a right triangle?

Answers

  1. 6 cm and 8 cm — The longest side 10 cm is the hypotenuse.
  2. 12 m — The leg is √(13² − 5²) = √144 = 12.
  3. No — 6² + 8² = 100, which is not 11² = 121.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources