March 7th, 2012, Aim: How do we find compund loci ???

A compound locus problem has two or more  locus conditions at the same time.  The different conditions in a compound locus problem are separated by the word "AND" or the words "AND ALSO".

( basically two problems put into one )

Example:

  A treasure is buried in your backyard.  The picture below shows your backyard which contains a stump, a teepee, and a tree.  The teepee is 8 feet from the stump and 18 feet from the tree.  The treasure is equidistant from the teepee and the tree AND ALSO 6 feet from the stump.  Locate all possible points of the buried treasure.

 

WHAT YOUR OUTCOME WOULD BE :

 

The red line represents the locus which is equidistant from the teepee and the tree (the perpendicular bisector of the segment).  The blue circle represents the locus which is 6 feet from the stump.  These two loci intersect in two locations.  The treasure could be buried at either "X" location.  

NOW YOU TRY IT !!!

1.

What is the number of points in a plane two units from a given line and three units from a given point on the line?

 Choose: [1] 1 [2] 2 [3] 3 [4] 4

2.

Two points  A and B are 12 units apart.  How many points are there that are equidistant from both A and B and also 6 units from B? Choose: [1] 0 [2] 1 [3] 2 [4] 3 *WRITE ANSWERS IN COMMENTS BELOW*                                       *Images and examples from regentsprep.org*

March 5th, 2012, Aim: How do we find the locus of points???

> The Locus is the set of all points that satisfy a given condition.

--There are different ways the locus of points can come out to be like:

1. The locus of points equidistant from a single point is a set of points equidistant from the point in every direction. 

*Example:

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: It comes out to be a circle with the original point at its center!

2. The locus of points equidistant from 2 points is the perpendicular bisector of the line segment determined with the 2 points.

*Example:

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

EASY HUH ? :)

 3. The locus points equidistant from a line are 2 lines on opposite sides equidistant and parallel to that line.

* Example:

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.

 

3. The locus of points equidistant from two intersecting lines are two additional lines that bisect the angles formed by the original lines (two intersecting lines half way between the two original lines.)

*Example: 

  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 .

ALWAYS REMEMBER : 

Remember:
A locus is just a set of points.

NOW YOU GIVE IT A TRY !!!

1.  Point A and point B are 26 miles apart. How many points are 20 miles from point A and 22 miles from point B?     

[1]  1     [2]  2     [3]  3   [4]  0

* WRITE ANSWER BELOW IN COMMENTS* 

*Pictures and animation from regentsprep.org*