Loci and constructions

A locus is a set of points that share a common property or satisfy a given condition. The plural is loci.

For example, consider a flat surface, like a piece of paper, and imagine a fixed point in the centre of this surface. Now, if you draw a set of points that are all at the same distance from this centre point, you will create a circle.

This circle represents a locus because all the points on the circle share the common property of being the same distance away (equidistant) from the centre point. In this case, each point is 3 cm away from point A.

In geometry, constructions refer to drawing shapes angles or lines accurately. To do this, we only use a pencil, a pair of compasses and a straightedge (e.g. a ruler).

Basic Geometric Constructions

Bisecting a line segment

Bisecting a line segment means dividing it into two equal parts or segments. To do this using a pair of compasses and straightedge, follow these steps:

1. Place the needle point of the compasses on one endpoint of the line segment (A), and open them to a width greater than half of the line segment’s length.

2. Draw two arcs above and below the line segment, ensuring that they cross the midpoint area.

3. Without changing the width of the compasses, move the needle point to the other endpoint of the line segment (B) and draw two more arcs.

  • These arcs should intersect the previous arcs, creating two intersection points above and below the line segment.

4. Using the straightedge, draw a straight line connecting the two intersection points. This line will bisect the line segment, creating two equal parts with the same length.

The point where the bisecting line intersects the original line segment is called the midpoint. The two new line segments have equal lengths, and their combined length is equal to the original line segment.

Perpendicular line through a point

You may be asked to construct a perpendicular line through a point that is not on the given line. To do this, follow these steps:

1. Place the needle point of the compasses on point P, and open them to a width greater than half the distance of line AB.

2. Draw an arc that intersects line AB at two points, one on each side of point P.

3. Without changing the width of the compasses, place the needle point on an intersection on line AB. Then draw an arc on the opposite side of the line, away from point P.

4. Again, without changing the width of the pair of compasses, repeat the previous step, placing the needle point on the second intersection. The arc formed should intersect the previous arc.

5. Using the straightedge, draw a straight line through point P and the two intersection points from steps 3 and 4.

  • This line is the perpendicular line to the given line AB, passing through point P

Angle bisector

An angle bisector is a line or ray that divides an angle into two equal angles. In other words, it cuts the original angle in half. To construct an angle bisector using a pair of compasses and a straightedge, follow these steps:

1. Identify the angle you want to bisect, which is formed by two rays or line segments with a common endpoint called the vertex.

2. Place the needle point of the pair of compasses on the vertex of the angle, and open the compasses to a suitable width. Draw an arc that intersects both rays or line segments that form the angle, creating two intersection points.

3. Without changing the width of the compasses, place the needle point on one intersection point, and draw an arc inside the angle.

4. Similarly, without changing the width of the compasses, place the needle point on the other intersection point, and draw another arc inside the angle. This arc should intersect the previous arc created in step 3, creating a new intersection point within the angle.

5. Using the straightedge, draw a straight line from the vertex through the intersection point created from steps 3 and 4.

  • This line will be the angle bisector, dividing the original angle into two equal angles

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