HONORS ASSIGNMENTS
1. Show that there is at least one non-constructible number in every interval [0, a],
a > 0.
2. Show that every circle centered at the origin contains at least two
non-constructible points.
3. Show that every circle contains at least two points which are non-constructible.
4. Define a similarity to be a transformation on the plane that preserves the between-
ness property of points and preserves angle measure. Prove that under a similarity, a
triangle is mapped to a similar triangle
5. Use Exercise 4. to show that if f is similarity, then there is a positive constant k
such that
kABBfAf
=
)()( for all segments
____
AB
.
6.
Consider points in the plane as ordered pairs (x, y) and consider the function on
the plane defined by ),(),( kykxyxf
=
, where k is a non-zero constant. Show that f is a
similarity.
7.
Let
l
r and
m
r be two reflections with lines of reflection l and m, respectively.
Show that the composition
'l
r
m
r
l
r
m
r=oo , where
'l is the reflection of l across m.
8.
Let
l
r be a reflection across l and T be a translation with displacement vector
parallel to l. Show that TrTr
ll
=
oo
9.
Show that if a rotation
d
iR
≠
has an invariant line, then it must be a rotation of
180 degrees.
10.
Suppose the composition of a rotation
d
iR
≠
with a reflection
1
r
is again a
reflection
2
r
, then
1
r
and
2
r
must pass through the center of rotation R.
