Loci Definition

Imagine a pilot flying an aeroplane at a certain speed, direction and altitude. By satisfying all these conditions, the plane had a good flight.

In Mathematics, a set of points that satisfy one or more conditions is called a locus. Let’s have an example: A circle with a centre point A and radius of 1 inch. To easily find the locus,

a. Make a drawing that satisfies the given conditions. This will help you describe the locus.
b. Describe the locus to check that all the conditions given are meet. Does every point satisfy the conditions?

The centre point A of the circle c with the radius of 1 inch around the circumference. To check if the condition has been satisfied, measure the radius. If it is 1 inch around the circumference and if at all points it is 1 inch from the radius, then it satisfies the condition.

Example 2
Describe the loci of points from a line that is 3 cm.

Let’s have a diagram and locate some points. First draw the line, then the points.

Description of the locus to determine if it satisfies the given condition. In this example, the loci of points can be two lines. Both of the lines are parallel to the given line. Since the measure of points around the lines are 3 cm, then it meets the given rules.

Three important loci
Locus is described as a set of a point that follows a certain condition. Here are loci that follow a rule that must have a given distance from a given point.

a. Circle the locus of points that are equidistant to the centre, given that the rules of a circle are that between the centre and the length of the radius must be equal in distance.

From the centre of the circle, the length of the radius is  3 cm. Therefore the circle c satisfies the condition of the point of a locus.

b. Perpendicular bisector the locus of points that are equidistant from two fixed points. From the word perpendicular and bisector, let’s make a description. Perpendicular refers to lines, rays or segments that intersect and form a 90-degree angle. The bisector is a line, segment or ray that cut into two equal parts.

The condition to satisfy the point of locus of a perpendicular bisector. First, it has a 90-degree angle and each segment is divided equally. Using a number line, it satisfies all conditions of the point of the locus. All the segments DA, EA, AC and BA measure 4 cm, therefore, they are congruent.

c. Angle bisector the locus of points that are equidistant from two lines. A line must cut an angle into two equal portions.

Example:
∠BAC bisected by segment AD. If ∠BAC is a right angle, therefore, ∠BAD and ∠DAC measures 45 degrees. Therefore the locus of the point is proven.