Expert: Sherry Wallin - 8/15/2010

Question
Hi,
I hope you can help me solve these problems:
I am supposed to describe in words the locus that satisfies the given condition.
(a)What is the locus of points in the plane of an angle that are equidistant from the sides of the angle/
(b)What is the locus of points in space that are equidistant from two parallel planes?
(c)What is the locus of points in a plane that are equidistant from points A and B in the plane?

2}How can I prove that tangents to a circle at the endpoints of a diameter are parallel? My question tells me to state what is given, what is to be proved, and my plan of proof.Then to write a two-column proof?I drew a diagram, but it is not helping much...
3}I also needed to prove that " If a diagonal of a parallelogram bisects an angle of the parallelogram, the parallelogram is a rhombus(I need to state my plan and give proof). How exactly am I supposed to prove these? Should I write theorems?Please let me know..
Thanks a million for helping!
Sara


Answer
Sara~

The first thing you need to know if what a locus is. It is just a location. A more specific definition is: A locus is a set of points that satisfy a given condition for a set of conditions.

So for part a the bisector of an angle will be the set of points that satisfies the points that are equidistant from the sides of the angle.

part b - talks about space and parallel planes so you need to be thinking about what set of points are equidistant from two parallel planes and that can only be the plane that is between two parallel planes that is halfway between the two parallel planes

part c - picture two points A and B and then try to figure out what set of points satisfies that they are equidistant. Since two points determine a line, find the midpoint of the line segment AB and draw a perpendicular and then when you choose a point X on the perpendicular there will be another point Y that is the same distance from the line segment AB.

2 - you are given a circle and all definitions of a circle (even though this is just implied) and you are given that you have a diameter and a tangent line where each end of the diameter touches the circle.  Another given, which is also implicit, is the definition of parallel. The usual definition of parallel is two lines in the same plane that never cross are parallel. The definition of tangent is a given that can be used, even though it isn't directly stated, you are told you have two tangent lines so you may use the definition of tangent. So a tangent is a line that intersects a circle at exactly one point. And finally you are given a diameter, so you can used the definition of diameter which is generally defined to be the longest chord of a circle and then this leads to being able to use the definition of a chord which is any line in a circle that extends from a side of a circle to another side of a circle without leaving the inside of a circle. One other fact you can use is that any tangnet to a circle is perpendicular to the line that connects the points of the chord of the circle. Now you have two lines that are perpendicular and their is a theorem that says two lines perpendicular to the same line are parallel.

I'm not sure how much of this you don't understand. A two column proof simply has two columns and on the left is the statements and on the right is the reason you can make that statement. So for beginners our two column proof might look like:

Statements                                  Reasons
1 we have a circle                    1  given
2 we have two tangents           2  given
3 we have diameter                 3  given
4 we have two tangents           4  given
5 we have two lines that          5  definition of a tangent
 intersect the circle at
 exactly one point
6 tangents intersect at            6 theorem that states the radius
 right angles                             (or any other line through the
                                                center of the circle) drawn to a
                                                tangent at the point of tangency   
                                                is perpendicular to the tangent at
                                                that point

etc...

3 - your premise or assumption is that you have a diagonal that bisects an angle of a parallelogram and what you need to prove is that the parallelogram is a rhombus. Here you get to use the definition of a rhombus (a quadrilateral with 4 equal sides) and what the definition of a parallelogram is which is a quadrilateral with two pairs of opposite sides parallel. This one is easy because you have a diagonals which will divide the parallelogram into 2 triangles and all you have to do is prove that two triangles are congruent and then use the fact that corresponding parts of congruent triangles are congruent. You will need to draw the other diagonal and do the same thing to show that all 4 sides are of equal length and then you get to conclude you have a rhombus since the definition of a rhombus is a parallelogram with 4 equal sides.

Math Prof