In these lessons, we will learn• the rules of the Locus theorem • exactly how the rule of the Locus Theorem have the right to be used in real human being examples.• exactly how to recognize the locus of clues that will certainly satisfy more than one condition.

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The adhering to diagrams give the locus that a suggest that satisfy some conditions. Scroll downthe page for more examples and solutions.


When a suggest moves in a airplane according to part given conditions the route along i beg your pardon itmoves is referred to as a locus. (Plural that locus isloci.).


A point P moves such the it is constantly m devices from the point Q.

Locus formed: A circle with center Q and radius m.


Example:Construct the locus of a point P in ~ a continuous distance of 2 centimeter from a fixed point Q.

Solution:Construct a one with center Q and also radius 2 cm.



A point P moves such that it is equidistant form two fixed pointsX and Y.

Locus formed: A perpendicular bisector the the heat XY.


Example:Construct the locus of suggest P relocating equidistant from resolved points X and Y and XY = 6 cm.

Solution:Construct a perpendicular bisector that the heat XY.



A point P move so the it is always m devices from a straight line AB.

Locus formed: A pair that parallel currently m unitsfrom AB.


Example:Construct the locus the a point P that moves a consistent distant of 2 centimeter from a directly line AB.

Solution:Construct a pair that parallel present 2 centimeter from AB.



A suggest P move so the it is always equidistant from twointersecting lines abdominal and CD.

Locus formed: edge bisectors of angles betweenlines abdominal muscle and CD.


Example:The following number shows two directly lines abdominal muscle and CD intersecting at allude O. Constructthe locus of point P such the it is always equidistant from ab and CD.


Example:Construct angles bisectors of angles between lines abdominal muscle and CD.


Five fundamental Locus Theorems and also How To usage Them

Locus organize 1: The locus of points in ~ a solved distance, d, from the point, p is a circlewith the given suggest P as its center and also d together its radius.Locus organize 2: The locus that the points in ~ a addressed distance, d, indigenous a line, l, is a pairof parallel present d street from l and also on either side of l.Locus to organize 3: The locus of points equidistant from 2 points, P and also Q, is theperpendicular bisector of the heat segment established by the 2 points.Locus to organize 4: The locus of clues equidistant from 2 parallel lines, l1and l2, is a line parallel to both l1 and also l2 and midwaybetween them.Locus theorem 5: The locus of point out equidistant from 2 intersecting lines, l1and l2, is a pair that bisectors that bisect the angles formed by l1and l2.

Example 1:A endowment map shows a treasure surprise in a park near a tree and also a statue. Themap suggests that the tree and the stature room 10 feet apart. The treasure is hidden 7 feetfrom the basic of the tree and additionally 5 feet indigenous the basic of the stature. How many places arepossible locations for the treasure to it is in buried? draw a chart of the sweetheart map, andindicate with an X each possible location the the treasure.

Example 2:The distance in between the parallel line l and m is 12 units. Allude A is on heat l.How countless points space equidistant from lines l and also m and 8 systems from suggest A.

Example 3:Maria’s backyard has actually two trees that room 40 feet apart. She wants to placelampposts so that the the write-ups are 30 feet from both of the trees. Attract a lay out to showwhere the lampposts might be inserted in relation to the trees. How plenty of locations because that thelampposts space possible?

Five rules Of Locus Theorem using Real world Examples

Locus is a collection of clues that meet a offered condition.There space five fundamental locus rules.Rule 1: given a point, the locus of points is a circle.Rule 2: offered two points, the locus of points is a straight line midway between the 2 points.Rule 3: provided a directly line, the locus of points is 2 parallel lines.Rule 4: provided two parallel lines, the locus of points is a heat midway in between the twoparallel lines.Rule 5: offered two intersecting lines, the locus of clues is a pair of currently that reduced theintersecting currently in half.

Intersection Of 2 Loci

Sometimes you might be compelled to determine the locus the a point that satisfies two or moreconditions. We could do this by creating the locus because that each that the conditions and also thendetermine wherein the two loci intersect.

Example:Given the line abdominal muscle and the suggest Q, discover one or an ext points that are 3 cm from ab and 5 cmfrom Q.


Solution:Construct a pair that parallel currently 3 centimeter from line AB. Draw a one with center Q andradius 5 cm.


The clues of intersections are shown by points X and also Y.

It way that the locus is composed of the 2 points X and also Y.

Example:Given a square PQRS through sides 3 cm. Construct the locus that a point which is 2 centimeter from Pand equidistant native PQ and also PS. Mark the points together A and also B.

Solution:Construct a one with center P and radius 2 cm. Due to the fact that PQRS is a square the diagonal line PRwould it is in the edge bisector that the angle formed by the lines PQ and also PS. The diagonal whenextended intersects the circle at points A and also B.


Note: A typical mistake is to identify only onepoint when there can be another suggest which can be found by expanding the constructionlines or arcs; as in the over examples.

GCSE Maths Exam inquiries - Loci, Locus and Intersecting Loci


Draw(i) the locus that a suggest that move so that it is constantly exactly 4 cm from the resolved pointX and(ii) the locus that points much less than 4 cm from the fixed allude X.

Draw the locus of point out no additional than 3 cm from A and no further than 4 cm from B.

Draw the locus that a allude exactly 3 cm away from right line AB.

Draw (i) the locus the a point equidistant indigenous the points X and Y.(ii) the locus of clues closer come the allude X than the allude Y.(iii) the locus of point out closer come X 보다 Y yet no less than 5 centimeter from X.

Draw the locus of clues closer come the line ab than the heat BC in the rectangle ABCD.

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A dog is on a lead tethered to a post in the corner of a garden. The lead is 5 m long.A cat is complimentary to roam all parts of the garden but is not enabled within 3 m the the houseby that owner. Show the for sure area that the cat can safely roam top top the chart below.

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