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 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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