Constructions, loci and geometric conditions · Extension
הוצג| English | משמעות |
|---|---|
| locus/ˈləʊkəs/ | מיקום |
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.
The method
- 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.
Which construction gives the locus of points equally distant from A and B?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
Equal distances to two POINTS define the perpendicular bisector of the segment joining them. Angle bisectors answer equal distances to two LINES - a different condition.
A worked case
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).
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Midpoint x-coordinate=(0+6)/2=3.
Where it goes wrong
- 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.
Not true: Points equally distant from two points lie on their angle bisector. Points equidistant from two points lie on the perpendicular bisector of the segment joining them; the angle bisector is equidistant from two lines.
Find the positive y-coordinate on x=3 and x²+y²=25.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
At x=3, y²=25-9=16. The positive y-coordinate is 4.
Points equidistant from two points lie on the perpendicular bisector of the segment joining them.
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
True. Points equidistant from two points lie on the perpendicular bisector of the segment joining them; the angle bisector is equidistant from two lines.
Applying it
- 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.
Find the distance from (3,4) to (0,0).
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Distance=√(3²+4²)=5.
Match each condition to its locus.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Each locus is the complete set of points satisfying its condition; compass arcs are the evidence of the construction method.
In your course
- OxfordAQA International GCSE Mathematics (9260), Extension tier, specification section 3.3.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Locus: The set of all points satisfying a stated geometric condition.
- Points equally distant from A and B lie on the perpendicular bisector of AB.
- Watch for: the perpendicular bisector concerns distance to two points; the angle bisector concerns distance to two lines.
Write from the line y equals 5
- The points are (0, 2) and (0, 8).
- Find the set of points equidistant from them, then one point on that line which is 4 units from (0, 5).
What is the set of points equidistant from (0, 2) and (0, 8)? Name one point on that set which is 4 units from (0, 5).
The average of 2 and 8 is 5. Moving 4 units horizontally from (0, 5) gives (4, 5).
What is the equidistant set?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
The two points are stacked vertically, so their perpendicular bisector is horizontal at y = 5.