Introduction to Loci: Definitions, Theorems and Examples

Loci is the plural form of locus. In Geometry, the locus is the set of points that follows certain properties or rules. Generally, the loci tell the construction of figures or shapes. For example, a circle is the loci of the points, which are at the same (fixed) distance from the fixed (same) point.
The word locus is originated from the word “location”. The place or entity where the points can be moved or located is termed as the geometrical shape before the 20th century. But, in modern Maths, the points with conditions or properties are considered entities.

Introduction to Loci

The word locus is originated from the word “location”. In Geometry, the locus is the set of points that follows certain properties or rules. Loci are the plural form of locus. Generally, the loci tell the construction of figures or shapes.
Thus, the locus is a path formed by the point that moves in a particular way or in certain conditions.

Let us explore this with a simple example:

The picture shown above gives the set of points of the headlight (white colour), and the set of points of tail lights (red colour) are traced by the condition that the vehicles follow the roadway path as shown.
Here, the set of points of the headlights or taillights follow the pathway of the vehicle. Thus, the set of points follow a certain rule. Thus, the set of points of lights shown gives the locus.
Set of points: Headlights or tail lights
Condition: Following the path of vehicles
Locus: The lane as shown in the figure.
In Latin, the word “locus” describes the “place”. Loci are the plural form of locus. Generally, the locus is usually drawn by using the dashes.

Examples of Loci

In Geometry, the locus is the set of points that follows certain properties or rules.

Here, the runner is running at a particular running track, which is formed with certain limits. Thus, the running path showed is the locus.

Example-2

The hands of a clock move round the clock, and they move at the same distance. The tip of the hour hand always moves at the same distance (equidistant) from the centre of the clock. Thus, the path moved by the hour hand creates the locus.

Important Theorems of Loci

There are mainly six important theorems in geometry that discuss the locus. Let us discuss these theorems in detail.

Theorem-1:

The locus formed by the set of points that are at the fixed distance “\(d\)” from the fixed point “\(P\)” is called the circle.
Here, the fixed point “\(P\)” is called the centre of a circle and the fixed distance “\(d\)” is called the radius of the circle.

In the above figure, the dotted line shown in the red colour describes the locus (circumference) formed by the set of points that are at a fixed distance (radius) from the fixed point (centre).

Example:

The path is formed by points that are located at the same distance from the one point-Circle.

Theorem-2:

The locus formed by the set of points that are at the fixed distance “\(d\)” from the given line “\(m\)” gives the pair of lines that are parallel to the given line. These two parallel lines are formed on either side of the given line.

In the above, the line shown in the red colour describes the parallel lines formed on either side of the line \(m.\)

Example:

The path is formed by the points that are located at the fixed distance from the one line – Parallel lines.

Theorem-3:

The locus formed by the set of points that are at the fixed (same) distance from the two points \(A\) and \(B\) of the line segment joining between these two points is the perpendicular bisector to that line segment.

In the above figure, the dotted line shown in the red colour describes the locus formed by the points \(A\) and \(B\) that are equidistant from two points of the line segment \(\overline {AB}.\)

Example:

The path is formed by the points that are located at the same distance from the endpoints of the line segment – perpendicular bisector.

Theorem-4:

The locus is formed by the set of points that are equidistant from the two parallel lines \({m_1}\) and \({m_2},\) is a line parallel to both the given lines \({m_1}\) and \({m_2}\) and it lies exactly halfway between the given lines.

In the above figure, the dotted line shown in the red colour describes the locus formed by the points that lie at the same distance from two parallel lines \({m_1}\) and \({m_2}.\)

Example:

The path is formed by the points that are located at the same distance between two parallel lines – The parallel line and lies halfway between those.

Theorem-5:

The locus formed by the set of points that are equidistant from the sides (arms) of the angle  and lies inside the angle gives the angular bisector of the given angle.

In the above figure, the dotted line shown in the red colour describes the locus \(\left( {BD} \right)\) formed by the points that are at the fixed distance from the sides \(\left( {AB,\,AC} \right)\) of the angle \(\angle ABC.\)
Here, the locus \(\left( {BD} \right)\) is the angular bisector of the \(\angle ABC.\)

Example:

The path formed by the points that are at the same distance – Angular bisector

Theorem-6:

The locus is formed by the set of points that are equidistant from the two intersecting lines \({m_1}\) and \({m_2}\) is the pair of lines and that bisects the angles formed by the given intersecting lines.

In the above figure, the dotted line shown in the red colour describes the locus formed by the set of points that are at a fixed distance from the two intersecting lines \({m_1}\) and \({m_2}\)

Example:

The path formed by the points that are at the same distance from the two intersecting lines – Pair of lined bisects the angles so formed.

Solved Examples – Introduction to Loci

Q.1. Construct a locus of the point \(P\) that is at the fixed distance \(2\,{\rm{cm}}\) from a fixed point \(Q.\)
Ans: We know that the locus formed by the set of points that are at the fixed distance “\(d\)” from the fixed point “\(Q\)” is called the circle.
Here, the fixed point “\(Q\)” is called the centre of a circle and the fixed distance “\(d\)” is called the radius of the circle.
Given, fixed distance is \(2\,{\rm{cm}}\)
\(d = 2\,{\rm{cm}}\)
Therefore, the locus of the point \(P,\) that is at the fixed point \(Q\) at the fixed distance \(2\,{\rm{cm}}\) is given below:

Q.2. Construct the locus of a point \(P\) moving at the same distance from the endpoints of the line segment \(XY.\)
Ans: We know that the locus formed by the set of points that are at the fixed (same) distance from the two points \(A\) and \(B\) of the line segment joining between these two points is the perpendicular bisector to that line segment.
Therefore, the locus of the point \(P\) moving at the same distance from the endpoints of the line segment \(XY\) passes through the midpoint of \(XY\) and is perpendicular bisector the line segment \(XY.\)
The locus is drawn shown below with a dotted line.

Q.3. Construct the locus of a point \(P\)  that moves at a fixed distance of  \(2\,{\rm{cm}}\) from the line \(AB.\)
Ans: The locus formed by the set of points that are at the fixed distance “\(d\)” from the given line “\(m\)” gives the pair of lines that are parallel to the given line. These two parallel lines formed on either side of the given line. Here, the locus is the pair of parallel lines (red in colour) to the given line \(m\) at the distance of \(2\,{\rm{cm}}\) on either side of the line \(m.\)

Q.4. Construct the locus of a point \(P\) that moves equidistantly from the two intersecting lines \(AB\) and \(CD\) as shown in the below figure.
Ans: Given the lines, \(AB\) and \(CD\) are intersecting lines.
The locus is formed by the set of points that are equidistant from the two intersecting lines \(AB\) and \(CD.\) It is the pair of lines, and that bisects the angles formed by the given intersecting lines.
The required loci are shown by red colour dotted lines.

The dotted lines shown in the above figure gives the locus of the intersecting lines.

Q.5. Construct a locus of a point \(P\) that is equidistant from the sides of the angle \(ABC,\) which is equal to \({60^o}.\)
Ans: Given that an angle \(ABC = {60^o}\)
We know that the locus formed by the set of points that are equidistant from the sides (arms) of the angle and lies inside the angle gives the angular bisector of the given angle.
The dotted line \(BD\) drawn from the vertex \(B\) of an angle \(ABC\) gives the locus of the given angle. And this line is equidistant from the sides \(AB\) and \(BC.\)

Summary

In this article, we have discussed the introduction and definition of the loci. We have studied the locus with the help of simple examples. This article also gives the six important theorems of the locus such as locus of the circle, locus of parallel lines, locus of angle bisector etc.
This article also gives examples that help us to construct the locus easily.

Frequently Asked Questions (FAQs) – Introduction to Loci

Q.1. What are the loci?
Ans: The plural form of locus is loci. The locus is the set of points that follows certain properties or rules.

Q.2. What is the locus of a circle?
Ans: The locus of the circle is the set of points that are at a fixed distance from the fixed point.

Q.3. What is the locus of a straight line?
Ans: The locus of a straight line is the collection of all the points that follow the same distances from two fixed points.

Q.4. What are the applications of loci?
Ans: The applications of the loci are circle, parabola, ellipse etc.

Q.5. What is the locus of the point maintaining the same distance from the endpoints of the line segment?
Ans: The locus of the point that is maintaining the same distance from the endpoints of the line segment is the perpendicular bisector of the given line.

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