In this lesson, we will learn

- the rules of the Locus Theorem
- how the rules of the Locus Theorem can be used in real world examples.
- how to determine the locus of points that will satisfy more than one condition.

When a point moves in a plane according to some given conditions the path along which it moves is called a **locus**. (Plural of locus is loci.).

** CONDITION 1** :

A point *P *moves such that it is always *m* units from the point *Q*

** Locus formed: ** A circle with centre Q and radius *m.*

* Example*

Construct the locus of a point * P * at a constant distance of 2 cm from a fixed point * Q. *

* Solution: *

Construct a circle with centre * Q* and radius 2 cm.

**
CONDITION 2** :

A point

** Locus formed: ** A perpendicular bisector of the line *XY*.

* Example: *

Construct the locus of point *P* moving equidistant from fixed points *X* and *Y* and *XY* = 6 cm.

* Solution: *

Construct a perpendicular bisector of the line XY.

**
CONDITION 3: **

A point

**Locus formed: **A pair of parallel lines *m* units from *AB*.

* Example: *

Construct the locus of a point *P* that moves a constant distant of 2 cm from a straight line *AB. *

* Solution: *

Construct a pair of parallel lines 2 cm from *AB. *

CONDITION 4:

A point *P *moves so that it is always equidistant from two intersecting lines *AB* and *CD*

**Locus formed: ** Angle bisectors of angles between lines *AB* and *CD*.

* Example: *

The following figure shows two straight lines *AB* and *CD* intersecting at point *O*. Construct the locus of point *P* such that it is always equidistant from *AB* and *CD. *

* Solution: *

Construct angles bisectors of angles between lines *AB* and *CD.*

The following video explains the five rules of locus theorem.

The following video will explain the five rules of locus theorem using real world examples.

Sometimes you may be required to determine the locus of a point that satisfies two or more conditions. We could do this by constructing the locus for each of the conditions and then determine where the two loci intersect.

* Example*

Given the line *AB* and the point *Q*, find one or more points that are 3 cm from *AB* and 5 cm from *Q*.

* Solution: *

Construct a pair of parallel lines 3 cm from line * AB. * Draw a circle with centre * Q* and radius 5 cm.

The points of intersections are indicated by points *X *and *Y*.

It means that the locus consists of the two points *X* and *Y. *

* Example: *

Given a square *PQRS* with sides 3 cm. Construct the locus of a point which is 2 cm from *P *and equidistant from *PQ* and *PS*. Mark the points as *A * and* B. *

* Solution: *

Construct a circle with centre *P* and radius 2 cm. Since *PQRS* is a square the diagonal *PR* would be the angle bisector of the angle formed by the lines *PQ* and *PS. *The diagonal when extended intersects the circle at points *A* and *B*

* Note:* A common mistake is to identify only one point when there could be another point which could be found by extending the construction lines or arcs; as in the above examples.

Intersecting Loci

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