Loci - Type 3
Loci 3
The locus of points which are equidistant from two given lines.
In plain terms:
There is a lot to explain here but if you follow all the steps in order then you'll have no trouble at all in working out how to construct this particular locus. You will need a compass and a ruler (and pencil!) to construct this loci type.
Follow these steps and look at the graphic below to follow what is happening during each step. I know it's a lot but it won't actually take you very long at all to draw this locus type.
1) Pick a random (but logical) start point to begin construction of the locus.
2) Draw two lines on diverging paths away from the start point so that an acute angle is created between them. These two lines will be your Given Lines.
3) From the start point use a compass to create one mark on each of the given lines you have drawn. These marks will be called your 1st Compass Mark and 2nd Compass Mark.
4) Your two compass marks (1st Compass Mark and 2nd Compass Mark) should be the exact same distance from the start point so it is vitally important that you do not adjust the compass setting between making your 1st and 2nd Compass Marks. Essentially what you're doing is drawing two small Arcs that are the same distance from the start point.
5) Using the 2nd Compass Mark as your new starting point, use the compass to draw a 3rd Compass Mark somewhere roughly in the middle between the two drawn lines.
6) Using the 1st Compass Mark as your new starting point (and without adjusting your compass from the last step - this is extremely important!) draw a 4th Compass Mark somewhere roughly in the middle between the two drawn lines. This 4th Compass Mark should cross over the 3rd Compass Mark and create an Intersect Point.
7) Draw a line from the original start point to the intersecting point of the 3rd and 4th Compass Marks.
This new line will be the Locus that is equidistant from the two given lines.
What you will have ended up with is a Locus that is actually an Angle Bisector. Both the angles that have been created by the locus from the start point will be identical.
A word of caution: do not draw any of the green dotted lines. I have put those in as guidance only! The green dotted lines are simply there to help you understand and learn how this particular Locus has been constructed.
If you are asked to draw this Locus in an examination then do not erase any of the compass marks you have drawn. The compass marks are the equivalent of showing your work.