Constructions, loci and geometric conditions · Extension · Constructions, lieux géométriques et conditions géométriques · Élargissement
| English | Français |
|---|---|
| locus/ˈləʊkəs/ | locus |
Where can both conditions hold?
- A router must be equally far from two rooms and within reach of a power point. Each condition creates a different set of possible positions.
- This lesson studies locus 轨迹: The set of all points satisfying a stated geometric condition.
Choose the mathematical structure
- Points equally distant from A and B lie on the perpendicular bisector of AB. Points at fixed distance r from C lie on a circle. Points equally distant from two intersecting lines lie on their angle bisectors.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines locus? · Quelle description définit correctement un lieu géométrique ?
The set of all points satisfying a stated geometric condition. · L'ensemble de tous les points satisfaisant une condition géométrique donnée.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For A=(0,0) and B=(6,0), the perpendicular bisector is x=3. Points also 5 units from A satisfy x²+y²=25. Substituting x=3 gives y²=16, so (3,4) and (3,-4) satisfy both conditions.
Constructions, loci and geometric conditions · Constructions, lieux géométriques et conditions géométriques
Points equally distant from A and B lie on the perpendicular bisector of AB · Les points équidistants de A et B se situent sur la médiatrice de AB
Explain why (3,4) belongs to both loci. · Expliquez pourquoi (3,4) appartient aux deux lieux géométriques.
Find the midpoint x-coordinate between (0,0) and (6,0). · Trouver l'abscisse du milieu entre (0,0) et (6,0).
Midpoint x-coordinate=(0+6)/2=3. · Abscisse du milieu=(0+6)/2=3.
Test a tempting shortcut
- The perpendicular bisector concerns distance to two points; the angle bisector concerns distance to two lines. A sketch is not a ruler-and-compass construction: preserve arcs as evidence of the method.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Points equally distant from two points always lie on their angle bisector. This claim is false. Explain which definition or assumption it violates.
Find the positive y-coordinate on x=3 and x²+y²=25. · Trouver l'ordonnée positive sur x=3 et x²+y²=25.
At x=3, y²=25-9=16. The positive y-coordinate is 4. · En x=3, y²=25-9=16. L'ordonnée positive est 4.
Points equally distant from two points always lie on their angle bisector. · Les points équidistants de deux points se situent toujours sur leur bissectrice.
The perpendicular bisector concerns distance to two points; the angle bisector concerns distance to two lines. A sketch is not a ruler-and-compass construction: preserve arcs as evidence of the method. · La médiatrice concerne la distance à deux points ; la bissectrice concerne la distance à deux droites. Un croquis n'est pas une construction au compas et à la règle : conservez les arcs comme preuve de la méthode.
Interpret a new situation
- Translate each condition into a locus before finding intersections. For a region closer to A than B, choose the correct side of the perpendicular bisector and show whether a boundary is allowed.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the distance from (3,4) to (0,0). · Trouver la distance entre (3,4) et (0,0).
Distance=√(3²+4²)=5.
Match each part of a complete solution to its purpose. · Associer chaque partie d'une solution complète à son but.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Une hypothèse justifie le modèle ; une vérification teste le résultat ; l'interprétation le relie à la question.
Use this in your course
- 9260 · Extension · 3.3. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The set of all points satisfying a stated geometric condition. Choose the relationship, show the method, check its assumptions and interpret the result.