WebLab STEM

Mathematics · 4. Geometry and measurement

From diagram to geometric proof

Use definitions, angle relations, similarity and Pythagoras to reason rather than measure a sketch.

40 minutesContent version: 2.1
Start the interactive pathwayOpen the virtual experiment

Driving question

What turns a suggestive diagram into a proof valid in every case?

Curriculum coverage

Vietnam Mathematics 6–10; Cambridge IGCSE Geometry and mensuration.

Model scope and limitations

Diagrams need not be to scale; screen measurement can suggest a conjecture but cannot replace proof.

Learning objectives

  • Use angle relations and triangle properties.
  • Apply Pythagoras correctly in right triangles.
  • Present a clear statement–reason chain.

Prerequisite knowledge

  • Squares, square roots and length units.

Core knowledge

Pythagoras

In a right triangle, the hypotenuse square equals the sum of the leg squares.

Similarity

Similar triangles have equal corresponding angles and proportional corresponding sides.

Proof

Each conclusion must follow from givens, definitions or known results, not appearance.

Worked example

A right triangle has legs 6 and 8: c² = 6²+8² = 100, so c = 10, taking the positive root for a length.

Misconceptions to avoid

  • The hypotenuse is opposite the right angle, not merely the side that looks longest.
  • AAA proves similarity, not congruence.

Virtual experiment procedure: Pythagoras theorem proof

  1. Mark givens and the target; do not add assumptions from appearance.
  2. Choose a theorem whose conditions hold and state those conditions.
  3. Check units, root signs and plausible lengths/angles.

Safety and cautions

  • In dynamic geometry, drag points through varied and near-degenerate cases before trusting a conjecture.

Evidence to collect

  • A solution with marked diagram, reasoning chain and correctly unitised conclusion.
Open the virtual experiment

Diagnostic check

1. Where is the hypotenuse?

  1. Opposite the right angle
  2. Adjacent to every acute angle
  3. Always horizontal
Show answer and explanation

Correct answer: Opposite the right angle

The hypotenuse is defined as the side opposite 90°.

2. Angles in a plane triangle sum to?

  1. 180°
  2. 360°
  3. 90°
Show answer and explanation

Correct answer: 180°

In plane Euclidean geometry, the sum is 180°.

Application practice

1. A right triangle has hypotenuse 13 and one leg 5. Other leg?

  1. 12
  2. 18
  3. 8
Show answer and explanation

Correct answer: 12

√(13²−5²) = √144 = 12.

2. Similar triangles have side scale factor 3. Area factor?

  1. 9
  2. 3
  3. 6
Show answer and explanation

Correct answer: 9

Area scales with the square of the side factor: 3² = 9.

3. Which condition can prove triangle congruence?

  1. Side–included angle–side
  2. Three angles
  3. One side
Show answer and explanation

Correct answer: Side–included angle–side

SAS fixes a triangle; AAA gives only similarity.

Academic references

  1. OpenStax Precalculus 2e — functions, geometry and probability

This lesson is maintained against the cited sources below. Simulations are learning models and do not replace supervised physical-laboratory safety procedures.

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WebLab STEM
Technical maintainer
Đức Tiến — Control & Automation Engineer