Loci and Construction

Loci and Construction

Loci Basics

A locus is a set of points that fulfills a specific condition.
Loci can be used in two-dimensional or three-dimensional contexts.
Loci can be singular (locus) or plural.

Drawing Loci

To draw the locus of a point that’s a fixed distance from another point, use a compass to draw a circle with the fixed point as the centre.
For a locus of points at a fixed distance from a line, draw the line and mark the distance along the perpendiculars.
For the locus of points equidistant from two other points, create perpendicular bisectors of the line segment.

Points of Intersection

Points where loci intersect each other fulfill the conditions of both loci*.
These intersection points may signify key parts of a geometric solution, e.g. the position from which an observer could see two objects.
Not all loci intersect at multiple points; some may not cross at all.

Constructing Loci

Construction involves using geometric tools to create shapes or solve problems.
Use a straight edge (like a ruler) for straight lines and a compass for circles and arcs.
Accuracy is crucial in construction; small errors can lead to large inaccuracies.

Bisectors and Perpendiculars

A bisector of an angle divides it into two equal smaller angles.
To construct a bisector between two lines, draw arcs from the point of intersection and connect their intersection.
A perpendicular to a line or plane is at a right angle (90 degrees) to it.
To construct a perpendicular, identify a point on the line and draw an arc centred on the point, then draw a second arc that cuts the first.

Triangle Construction

To construct a triangle, you must know at least three elements: typically a combination of sides and angles.
Draw one side first, then use a compass to construct the other two from its ends.
Check your final triangle to ensure it fulfills the initial conditions.

Practical Applications

Loci and construction are used in many real-world situations, including navigation, building and design, and visualizing mathematical situations.
Knowing how to draw and interpret them can help in understanding and solving complex problems.
Always follow the required steps for your specific calculation, and remember to check your work for accuracy.