ریاضی
Locus and Loci: Points That Obey a Rule
A locus is every point that obeys a rule. The four that exam papers use — circle, racetrack, perpendicular bisector, angle bisector — and how regions differ.
Locus
The set of all points that satisfy a given rule about distance or position, drawn as the path or region those points form.
- دیگر نام
- Loci, the plural
- طلبہ کہاں پڑھتے ہیں
- Grade 9 or 10 mathematics, in the constructions and loci topic of GCSE and IGCSE geometry.
مختصر جواب
A locus is the set of every point that satisfies a given rule, drawn as the path or region those points form. The locus of points 3 cm from a fixed point is a circle of radius 3 cm; the locus of points equidistant from two fixed points is their perpendicular bisector.
ایک مثال
Locus of points exactly 3 cm from O → a circle, centre O, radius 3 cm
Every point on that circle is 3 cm from O and every point 3 cm from O is on it, which is what makes it the locus rather than merely part of it. Change the wording to less than 3 cm and the answer becomes the shaded inside of the circle instead of the curve itself.
The four loci a paper actually asks for
Loci questions look varied because they are dressed up as gardens, fields and mobile phone masts, but almost all of them are built from four basic rules. Learn what shape each produces and most questions become a matter of translating the story.
Each of these is drawn with compasses and a straight edge, not sketched by eye, because the accuracy of the boundary is what is being marked.
- A fixed distance from a point — a circle with that radius
- A fixed distance from a straight line segment — two parallel lines either side, joined round the ends by semicircles, giving a running-track shape
- Equidistant from two points — the perpendicular bisector of the segment joining them
- Equidistant from two lines meeting at a point — the bisector of the angle between them
From a path to a region
The wording decides whether the answer is a line or an area. Exactly 5 m from the wall is a line; within 5 m of the wall is everything between that line and the wall, and it is shaded. Nearer to A than to B is the half of the diagram on A's side of the perpendicular bisector.
Most full-mark questions combine two or three conditions and ask for the region satisfying all of them. Construct each boundary separately, in pencil, then shade only the overlap. Working out the boundaries one at a time is what keeps the diagram readable.
Why it has to be constructed
A locus question is a constructions question wearing a story, and the marks follow the same rules. Compasses for arcs, a sharp pencil, and the construction arcs left visible on the diagram.
The word itself is Latin for place, and its plural is loci — so one rule gives a locus and several rules give several loci. It is the only part of school geometry where the Latin plural is still standard.
عام سوالات
Is a locus a line or a region?
Either, depending on the rule. A rule using exactly gives a path — a circle, a straight line, an arc. A rule using less than, within, or at most gives a region, drawn by constructing the boundary first and then shading one side of it. Read the wording before drawing anything.
What is the locus of points 2 cm from a 6 cm line segment?
A running-track shape. Two straight lines 6 cm long sit 2 cm above and below the segment, and each end is capped by a semicircle of radius 2 cm centred on the endpoint. The semicircles are the part most often forgotten, and they are where the marks usually are.
How do I show the region nearer to one point than another?
Construct the perpendicular bisector of the segment joining the two points, then shade the side containing the point you want to be near. Every point on that side is closer to it, every point on the other side is closer to the other, and the bisector itself is exactly equidistant.
Do I have to use compasses, or can I measure with a ruler?
If the question says using ruler and compasses, then compasses, with the arcs left showing. Where it does not say, an accurately measured construction may be accepted, but the arcs cost nothing and remove the argument. A freehand sketch of a boundary is not accepted either way.
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