Year 8 Mathematics · AC9M8M06

Pythagoras’ Theorem

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Identify the right angle and hypotenuse first, then use the…

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Key conceptTeach from the board

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.

Worked examplesWe do

Worked examples

AC9M8M06 - Pythagoras’ Theorem
Example 1

Label c on three differently oriented right triangles.

Example 2

Find c for legs 9 and 12.

Example 3

Find a missing leg when c=17 and the other leg=8.

Example 4

Test 8,15,17 as a Pythagorean triple.

Curriculum examplesCopied content

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.

LessonAC v9VictoriaNSW
Pythagoras’ theoremAC9M8M06VC2M8M06 — ExactMA4-PYT-C-01; MAO-WM-01 — Exact
Questions and answersWith answers
  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.
Practice and reviewReady for practice
  • 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.
Curriculum alignmentStart here
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