What should you draw, and what should remain unknown?
FNEGE includes elementary geometry in the Calculation section. This workshop is limited to rectangles, right triangles, and rectangular prisms; it is not a guaranteed list of test questions. The diagrams described are schematic and not to scale.
Start by labeling the vertices, then write in the given lengths. A right angle must be stated in the problem or otherwise justified; two segments that look the same are not necessarily equal. Mark the unknown with a question mark. This discipline helps you avoid unconsciously adding symmetry and solving an easier problem than the one given.
Description of our example rectangle: A is at the upper left, B at the upper right, C at the lower right, and D at the lower left. Sides AB and CD are horizontal; BC and AD are vertical. The ▭ symbol is a visual cue, but the labels and written dimensions provide the usable information.
| Figure or quantity | Relationship |
|---|---|
| Rectangle with sides L and l | Area = L × l; perimeter = 2(L + l) |
| Triangle with base b and corresponding height h | Area = b × h / 2 |
| Right triangle with perpendicular sides a and b and hypotenuse c | c² = a² + b² |
| Rectangular prism with dimensions L, l, h | Volume = L × l × h |
Why distinguish area from perimeter?
Imagine paint covering an entire panel and a border running around its edge: these two quantities are not measured with the same formula.
Une plaque rectangulaire mesure 12 cm de long et 5 cm de large. On double uniquement sa longueur. Quelle est sa nouvelle aire ?
Choose an answer.
The solution, step by step
120 cm².
The method in action
- The new length is 24 cm and the width remains 5 cm. The area is 24 × 5 = 120 cm².
- The original area was 60 cm²; doubling one dimension doubles it. Doubling both dimensions would multiply it by four.
- 34 is the original perimeter and 58 is the new perimeter; their unit would be centimeters, not square centimeters. 240 would correspond to doubling both dimensions.
How do you identify the terms in the Pythagorean theorem?
Reconstruct the diagram: A is at the right angle; B is to the right of A and C is above A. Segments AB and AC are the perpendicular sides. BC connects their endpoints and is the hypotenuse, opposite the right angle.
Le triangle ABC est rectangle en A, avec AB = 9 cm et AC = 12 cm. Combien mesure BC ?
Choose an answer.
The solution, step by step
15 cm.
The method in action
- BC² = AB² + AC² = 81 + 144 = 225. Therefore, the positive length BC is 15 cm.
- 225 is the square of the length, not the length itself. 21 adds the two sides, measuring a route through A rather than segment BC.
- The result must be greater than 12 cm because the hypotenuse is the longest side, but less than 9 + 12 = 21 cm. This check confirms that 15 is reasonable.
The pitfall
Trusting the apparent size of a diagram
A nearly square rectangle does not let you assume L = l. A nearly horizontal diagonal does not mean the angle is zero. For each relationship, ask which statement or property justifies it. If the only answer is “you can see it,” you are missing a proof.
Use the same care with conversions: 1 m² equals 100 × 100 cm², or 10,000 cm². The factor of 100 applies to a length. For area or volume, it applies to each dimension.
How can you check the dimensions of a result?
Calculate the volume of a rectangular prism after expressing all three dimensions in the same unit.
Un bac en forme de pavé droit a pour dimensions intérieures 50 cm, 40 cm et 30 cm. Quel est son volume en litres, sachant que 1 litre = 1 000 cm³ ?
Choose an answer.
The solution, step by step
60 liters.
The method in action
- The volume is 50 × 40 × 30 = 60,000 cm³. Divide by 1,000 to get 60 liters.
- 120 comes from adding the dimensions and does not represent a volume. 6 and 600 indicate a factor-of-ten error.
- Check in decimeters: 5 × 4 × 3 = 60 dm³. Since one liter equals one dm³, this confirms the result.
Key takeaways
- Use only properties that are given or proven.
- Distinguish perimeter, area, and volume before choosing a formula.
- Check the unit and order of magnitude.