Constructions, loci and geometric conditions · Extension
| English | 中文 | Pinyin |
|---|---|---|
| locus/ˈləʊkəs/ | 轨迹 | guǐ jì |
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?
The set of all points satisfying a stated geometric condition.
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
Points equally distant from A and B lie on the perpendicular bisector of AB
Explain why (3,4) belongs to both loci.
Find the midpoint x-coordinate between (0,0) and (6,0).
Midpoint x-coordinate=(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.
At x=3, y²=25-9=16. The positive y-coordinate is 4.
Points equally distant from two points always lie on their angle bisector.
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.
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).
Distance=√(3²+4²)=5.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the 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.