Chapter in a nutshell: A locus is the path traced by a point that moves according to a given condition. The chapter is about recognising the standard loci (a circle, a perpendicular bisector, angle bisectors, parallel lines) and constructing them with compass and ruler.
1. Key Idea
- Locus: the set of all points (and only those points) that satisfy a given geometric condition. Every point on the locus obeys the rule; every point obeying the rule lies on the locus.
2. Standard Loci (the "formula" set)
| Condition on a moving point | Locus |
|---|---|
| At a fixed distance $r$ from a fixed point O | a circle of radius $r$ centred at O |
| Equidistant from two fixed points A and B | the perpendicular bisector of AB |
| Equidistant from two intersecting lines | the pair of angle bisectors of the angles between them |
| At a fixed distance $d$ from a given line | two lines parallel to it, at distance $d$ on each side |
| Equidistant from two parallel lines | the line midway (parallel) between them |
| Such that it subtends a fixed angle on a fixed segment | an arc of a circle through the endpoints |
3. Useful Theorems (justify the loci)
- Perpendicular bisector: any point on it is equidistant from the two endpoints (and conversely).
- Angle bisector: any point on it is equidistant from the two arms of the angle (and conversely).
4. Construction Notes (method)
- To draw the locus equidistant from A and B โ construct the perpendicular bisector of AB.
- To draw the locus equidistant from two lines โ construct the angle bisector(s).
- The required point is usually the intersection of two loci (e.g. "equidistant from A and B and at distance d from line $l$").
5. Prerequisite Skills
- Constructions: perpendicular bisector, angle bisector, drawing circles/arcs with a compass.
6. Worked Example (one, for the method)
Q. Describe the locus of a point that is always 4 cm from a fixed point P. Solution: A circle of radius 4 cm with centre P.7. Common Mistakes to Avoid
- Giving only some points instead of the whole path (a locus is the complete set).
- Confusing "equidistant from two points" (perpendicular bisector) with "equidistant from two lines" (angle bisector).
- Forgetting a fixed distance from a line gives two parallel lines (both sides).
- Not taking the intersection of two loci when two conditions are given.
8. Likely Exam Questions (with crisp answers)
- Define a locus. โ The path of a point moving under a given condition (all points satisfying it).
- Locus of points at fixed distance from a point? โ A circle.
- Locus equidistant from two fixed points? โ The perpendicular bisector of the segment joining them.
- Locus equidistant from two intersecting lines? โ The bisectors of the angles between them.
- Locus at a fixed distance from a straight line? โ Two parallel lines, one on each side.
- Locus equidistant from two parallel lines? โ The parallel line halfway between them.
- What property has every point on a perpendicular bisector? โ It is equidistant from the two endpoints.
- How is a point satisfying two conditions found? โ As the intersection of the two loci.
- Locus equidistant from the arms of an angle? โ The angle bisector.
- Is a single point a locus? โ It can be (e.g. the intersection of two loci).
Extended & Prerequisite Formulas (full)
- Fixed distance r from a point โ circle radius r; equidistant from 2 points โ perpendicular bisector; equidistant from 2 lines โ angle bisectors; fixed distance d from a line โ 2 parallel lines.
- Prereq: perpendicular bisector & angle bisector constructions; distance formula; circle (2ฯr, ฯrยฒ).