loci

Aim: How do we find locus of points?
By finding what satisfies the problem and inorder to do that here are some theorems I want to show you.

By following the different types of loci problems which are the following:\

A fixed distance from a point.

1.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.

Hint: If you are ever asked to find the locus of points from a point it is always a circle!
2. The locus theorom number two.                                                                                                            

The locus of points at a fixed distance, d, from a line, l, is a pair of parallel lines d distance from l and on either side of l.

The locus of points equidistant from two lines is 1 line and they are always parallel.

3. The third theorem is the following:                                    

 

The locus of points equidistant from two points, P and Q, is the perpendicular bisector of the line segment determined by the two points.

This picture shows that all the points equidistant from two other points is a perpendiculsr bisector.

4. The fourth theorem is the following: this is an example from regents pre.org!!

 During your morning jog, you run down an alley between two buildings which are parallel to one another and are 20 feet apart.  Describe your path through the alley so that you are always the same distance from both buildings.

Answer:

To maintain an equal distance from each building, you must jog in a straight line parallel to the buildings and halfway between them.  In this problem, since the buildings are 20 feet apart, you will jog on a line 10 feet from each building. 

The locus of points equidistant from two parallel lines, l1 and l2 , is a line parallel to both l1 and  l2 and midway between them.

 5. The final theorem is the following:

The locus of points equidistant from two intersecting lines, l₁ and l₂, is a pair of bisectors that bisect the angles formed by l₁ and l₂ .

 In this picture line 1 and line 2 show all the possible points that are equidistant to the x and y axis which is the red dotted lines!!



Try it out!!
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?

 ALL PICTURES ARE FROM REGENTSPREP.ORG i LUV U REGENTS PREP!!