Locus Of A Moving Point

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Loci In Geometry
Loci
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The following diagrams give the locus of a point that satisfy some conditions. Scroll down the page for more examples and solutions.

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 center 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 center Q and radius 2 cm.

CONDITION 2:

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

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 P moves so that it is always m units from a straight line AB.

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.

Example:
Construct angles bisectors of angles between lines AB and CD.

Five Fundamental Locus Theorems And How To Use Them

Locus Theorem 1: The locus of points at a fixed distance, d, from the point, P is a circle with the given point P as its center and d as its radius.
Locus Theorem 2: The locus of the points at a fixed distance, d, from a line, l, is a pair of parallel lines d distance from l and on either side of l.
Locus Theorem 3: The locus of points equidistant from two points, P and Q, is the perpendicular bisector of the line segment determined by the two points.
Locus Theorem 4: The locus of points equidistant from two parallel lines, l₁ and l₂, is a line parallel to both l₁ and l₂ and midway between them.
Locus Theorem 5: The locus of points equidistant from two intersecting lines, l₁ and l₂, is a pair of bisectors that bisect the angles formed by l₁ and l₂.

Example 1:
A treasure map shows a treasure hidden in a park near a tree and a statue. The map indicates that the tree and the stature are 10 feet apart. The treasure is buried 7 feet from the base of the tree and also 5 feet from the base of the stature. How many places are possible locations for the treasure to be buried? Draw a diagram of the treasure map, and indicate with an X each possible location of the treasure.

Example 2:
The distance between the parallel line l and m is 12 units. Point A is on line l. How many points are equidistant from lines l and m and 8 units from point A.

Example 3:
Maria’s backyard has two trees that are 40 feet apart. She wants to place lampposts so that the the posts are 30 feet from both of the trees. Draw a sketch to show where the lampposts could be placed in relation to the trees. How many locations for the lampposts are possible?

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Five Rules Of Locus Theorem Using Real World Examples

Locus is a set of points that satisfy a given condition.
There are five fundamental locus rules.
Rule 1: Given a point, the locus of points is a circle.
Rule 2: Given two points, the locus of points is a straight line midway between the two points.
Rule 3: Given a straight line, the locus of points is two parallel lines.
Rule 4: Given two parallel lines, the locus of points is a line midway between the two parallel lines.
Rule 5: Given two intersecting lines, the locus of points is a pair of lines that cut the intersecting lines in half.

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Intersection Of Two Loci

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 center 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 center 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.

GCSE Maths Exam Questions - Loci, Locus And Intersecting Loci

Examples:

1. Draw
   (i) the locus of a point that moves so that it is always exactly 4 cm from the fixed point X and
   (ii) the locus of points less than 4 cm from the fixed point X.

2. Draw the locus of points no further than 3 cm from A and no further than 4 cm from B.

3. Draw the locus of a point exactly 3 cm away from straight line AB.

4. Draw
   (i) the locus of a point equidistant from the points X and Y.
   (ii) the locus of points closer to the point X than the point Y.
   (iii) the locus of points closer to X than Y but no less than 5 cm from X.

5. Draw the locus of points closer to the line AB than the line BC in the rectangle ABCD.

6. 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 free to roam all parts of the garden but is not allowed within 3 m of the house by its owner. Show the safe area that the cat can safely roam on the diagram below.

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