Pythagoras’ Theorem — AC9M8M06

• Identify the hypotenuse in a right-angled triangle.
• Use a²+b²=c² to find missing side lengths.
• Test whether three side lengths form a right triangle.
• Apply the theorem in practical distance and design problems.

Recall square numbers, square roots, right angles, triangle side notation, substitution and rearranging a simple equation.

The theorem applies only to right-angled triangles. The hypotenuse c is opposite the right angle and is always the longest side. Then a²+b²=c².

To find a shorter side, subtract a known leg-square from the hypotenuse-square before taking the positive square root. To test a triangle, use the largest side as the candidate c and compare a²+b² with c².

Preserve the full AC9 elaboration range: compare historical demonstrations/proofs from several mathematical traditions, solve practical missing-length problems, examine how square relationships differ for acute/right/obtuse triangles, recognise Pythagorean triples, and connect right-triangle distance to coordinate-plane/predictive navigation contexts without drifting into later trigonometry.

• Choosing any side as c. c must be opposite the right angle and longest.
• Adding side lengths instead of squares. The theorem relates squared lengths.
• Using a²+b²=c² for a non-right triangle. The right-angle condition is essential.
• Keeping both ± square roots for a length. Physical side length uses the positive root.

1. Label c on three differently oriented right triangles.
2. Find c for legs 9 and 12.
3. Find a missing leg when c=17 and the other leg=8.
4. Test 8,15,17 as a Pythagorean triple.

1. Find the hypotenuse for legs 5 and 11.
2. Find a leg when c=20 and the other leg=16.
3. Test whether 6,8,11 forms a right triangle.
4. Find the diagonal of a 9 m×12 m rectangle.
5. Find the straight-line distance between (1,2) and (7,10).
6. Explain why 3,4,5 and 6,8,10 are related triples.
7. Describe one historical or practical way to demonstrate the theorem’s relationship.

A builder measures a rectangular frame as 2.4 m by 3.2 m and one diagonal as 4.05 m. The theoretical right-rectangle diagonal is 4.0 m. Explain what the discrepancy could mean, what measurement accuracy matters, and how a second diagonal would help diagnose whether the frame is square.

1. How do you identify the hypotenuse? It is opposite the right angle and is the longest side.
2. When does Pythagoras apply? Only for right-angled triangles.
3. How do you test a triangle from side lengths? Square the two shorter sides and compare their sum with the square of the longest.
4. What is a Pythagorean triple? Three integer side lengths satisfying a²+b²=c².

1. A ladder reaches 4.8 m up a wall and its foot is 2 m from the wall. Find ladder length to 2 decimal places.
   The ladder is the hypotenuse.
2. A triangle has sides 9,12,15. Determine whether it is right-angled and justify.
   Use 15 as c and compare squares.
3. Find the distance between A(−2,3) and B(4,11).
   Use horizontal/vertical differences as perpendicular legs.

• I identify c correctly.
• I find hypotenuse or leg lengths.
• I test right-triangle side sets.
• I apply the theorem to coordinates/practical geometry.

Exit ticket: Why must the longest side be tested as c when checking three lengths?

Australian Curriculum v9.0 — AC9M8M06: use Pythagoras’ theorem to solve problems involving the side lengths of right-angled triangles.

Victorian Curriculum F–10 Version 2.0 — Level 8, VC2M8M06: Exact.

NSW Mathematics K–10 Syllabus (2022) — Stage 4, MA4-PYT-C-01; MAO-WM-01: Exact for applying Pythagoras’ theorem in contexts, supported by Working mathematically.

• AC9M8N01 — irrational square roots
• AC9M8SP01 — congruence and similarity

• Australian Curriculum v9.0
• Victorian Curriculum F–10 Version 2.0 Mathematics
• NSW Mathematics K–10 Syllabus (2022)

Curriculum equivalents for Pythagoras’ theorem to solve problems involving the side lengths of...

Mapped skill: use Pythagoras’ theorem to solve problems involving the side lengths of right-angled triangles

These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.

Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.