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Consider:
You are standing in the middle of your classroom. The teacher asks
Mike, your classmate, to stand 5 feet away from you. The teacher
then asks Carlos, Ashley and Anita to also stand 5 feet away from you.
Can you start to picture what is happening? If all of your
classmates were to stand 5 feet away from you, what geometric shape or
path would
your classmates be forming?
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Answer: The students would be forming a
circle around you. You are the center of the circle, and the radius
of the circle would be a fixed distance of 5 feet.
The path where the students are standing is a locus.
The locus in this situation is a circle.
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Stated formally, this path or shape
becomes our first locus theorem:
| Locus
Theorem 1: (point) |
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The locus of points
at a fixed distance, d, from point P is a circle
with the given point P as its center and d as its
radius.
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Consider:
Niko gets a stone wedged in the
tire of his dirt bike. He lifts the bike and spins the
wheel to look for the stone. Describe the path of the
stone as Niko spins the wheel?
The stone is following a circular
path around the center of the tire, illustrating our theorem.
Ponder:
If Niko "pushed" the bike along the ground to search for the
stone, would the path of the stone still be a circle?
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No. The
path is no longer circular, as the wheel rolls forward.
Remember we are following the path of the stone in each of its
possible locations. The path would resemble:
If you follow the path of a fixed
point on a circle as it rolls on a line, it forms a curve called
a cycloid. |
Consider: When he is not in the house,
Fido is tied to a stake in the backyard. His leash, attached to
the stake, is 15 feet long. When traveling at the end of his
leash, what is the locus of Fido's path?
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Point
P,
from the theorem,
is the stake to which Fido, the dog, is tied. His
leash is 15 feet long. The path that Fido can travel
at the end of his leash is "the locus of
points".
The locus
of points at a distance of 15 feet from point P is a
circle (with center P and radius 15). |
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