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Lesson 148: Intro to Locus Problems
The locus is a set of points that satisfy a certain condition. In this lesson we'll learn the basics of how this topic works.
The definition of locus (plural: loci) is the set of all points that satisfy one or more conditions. The concept is best explained by example:
What
is the locus of points that are 5 cm from a given point P?
Make a point near the center of your paper, and label it P.
Using a ruler, measure 5 cm away from that point, in the direction
of your choice, and plot a point there. Now do the same thing,
but measure in a different direction starting from the point P.
If you keep doing this, you'll see that your points form a circle. See the diagram at left. We can say that the locus of points that are 5cm from point P is a circle with a radius of 5cm, centered at point P. Keep thinking about that until it makes sense to you.
That is the first basic locus. The set of points that are a distance r from a point P is a circle centered at P, with a radius of r. Note that in all of these diagrams, the actual locus will be represented with a dashed line.
The
next common locus is the set of points at a fixed distance d from a
line L. For example, what is the locus of points that are 5cm
from line L. First draw line L. The line can be at any angle,
but drawing it horizontally will make it easier for you. Let's
measure 5cm starting at the line measuring straight up. Plot the point.
If you keep doing this in different locations on the line, you'll
see that what you actually form is another line that is 5cm above
line L. We can also do the same thing below the line.
Each of these lines is parallel to line L, and 5cm away from it.
See the diagram at left. Again, keep thinking about this until
it makes sense.
There
are three other common loci. Let's look at the locus of points
that are equidistant from two points A and B. This means we
need points that are the same distance from A as they are from B.
One such point will lie directly between them. But, we can
also plot points that are above and below that point, and they will
also be equidistant from A and B. The locus is actually the
perpendicular bisector of line segment AB. See the diagram at
left.
Another
common locus is the locus of points that are equidistant from two
parallel lines. That is pretty easy to visualize. It is
a line in the between those two lines, parallel to each of them, and
midway between them. See diagram at left.
The
final kind of locus is the set of points equidistant from two
intersecting lines. The locus will be the bisectors of each
pair of vertical angles formed by the lines. This last one is
a bit tricky. Study the diagram at left.
Many problems will ask you to simply draw the locus, but some problems ask you to find the set of points that satisfy two different types of loci at the same time. There are many different variants of such problems. Just sketch out each locus, and then look for the points where the two loci intersect. Very often they will only intersect at two points. Sometimes the question only asks how many such points there are, and your sketch is not evaluated, but in some cases it is. Always just read the question very carefully and take it slow.
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