1. Find the distance between P and 
. (That is, construct the segment whose length is the
distance between P and 
.)
- What
is true about P and the endpoints of the segment you constructed from the
intersection of the circle centered at P and intersecting
?
- What is true about P and the segment bisector?
- Why did you need to construct a segment bisector at all?
- In what ways did our hypothesis about perpendicular bisectors help in solving this problem?
Use the statement below for problem 2.
The distance between a point and a segment is defined to
be the distance between that point and the line containing that segment.
2.
With the above definition in mind, what strategy would you use
to find the distance between H and 
.
Strategy (it doesnÕt have to be complicated):
3. Define the distance between a point and a ray so that it is consistent with definitions of the distance between a point and a line and the distance between a point and a line segment.
Given point A and 
, the distance between A and 
is defined to be
4.
Using your definition, what strategy would you use to find the
distance between K and 
.
Strategy:
5. Given angle ABC, construct a point that is equidistant from BOTH ray BA and ray BC.
a.
Strategy Part I:
Use these questions to understand the kinds of things you need to think
about when you hear the word ÔstrategyÕ.
A strategy is not just a list of steps; you can use a strategy to help solve more complicated
problems (like this one).
i)
What do you think will be true about a point that is
equidistant from two rays?
ii)
Could there be more than one point that is equidistant from
two rays?
iii)
Now imagine the set of all points equidistant from two
raysÉwhat does this set of points look like itÕs ÔdoingÕ to the angle (the word
ÔdoingÕ isnÕt exactly accurate, but itÕll meet our purposes for now)?
iv)
A construction with the properties above is called an angle
bisector. So, what shall we mean we sayÉ Ôthe angle bisector of angle
ABCÕÉ? (Here is where you provide
a definition. J)
b.
Strategy Part II:
OK, so we got sidetracked with our definition of angle bisector, now
back to the problemÉ
i)
WeÕve talked about how the points on an angle bisector and the
sides of our angle relate to one another.
Now imagine just ONE point on the angle bisector. Also remember our defintion of Ôthe
distance between a point and a lineÕ.
How do we measure the distance of a point on the angle bisector to the
sides fo the angle?
ii)
What must be true about those intersection points in relation
to the vertex of angle ABC?
iii)
So how can we construct two points on 
and 
that are each
equidistant from the vertex?
iv)
Now how can we construct a point or points that is/are
equidistant from both sides of 
and 
?
c.
NOW construct the angle bisector of angle ABCÉ
