# Constructions, Loci

## Introduction

Knowing how to construct lines/triangles aren't as important these days but it's still good to know if you don't have any other equipment around. Loci can be used to find maximum area of animals grazing for example, so in this sense, loci methods are probably still used today.

## Constructions

Perpendicular/Line Bisector - this can be used to form a 90° angle

1) Open the compass to about $\dfrac{3}{4}$ of the length of the line

2) Put the compass point at one end of the line and draw an arc above and below

3) Put the compass point at the other end of the line keeping the compass open at the same distance

4) Draw another two arcs which cross your original ones

5) Use a ruler to join the two points where the arcs have crossed above and below the lines

6) You should have a line that crosses your line in the middle forming two right angles

Angle Bisector - this can be used to bisect the perpendicular bisector to form a 45° angle

1) Open the compass to about two thirds of the length of the line

2) Place the point of the compass on the vertex of the angle

3) Draw two arcs, one on each line

4) Leave the compass open the same distance and put the point on one of the arcs, draw another arc in line with the angle and repeat with the other line

5) Use a ruler to join the vertex of the angle and the point where the arcs you have just drawn cross

6) you should have a line that splits your angle in two

Equilateral triangle - this can be used to form a 60° angle

1) Open the compass to the length of your line

2) Put the compass point at one end of your line

3) Draw an arc above the line

4) Put the compass point at the other end of your line and draw an arc that crosses the previous one

5) Use a ruler to join the point where the arcs cross to each end of your line

6) You should now have an equilateral triangle and three 60° angles

## Loci

Loci are the path drawn out when following a certain rule (for example, $1$cm away from a line). So in the cases below every line is a certain distance away from the line/point, but it's always the same distance away.

For this locus the compass is opened up to $5$cm to draw a path that is consistently $5$cm away from the point.

For the majority of the line the ruler is used but at the ends a compass is used to keep the path the same distance away.

This one is mostly similar to the above but a square is needed at one point to keep the path consistent.

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