Homework 4 (not for credit)
1. Suppose the lines
‘,m,n
have equations
X
= 2,
Y
= 3, and
Y
= 5. Find the equations
for
σ
m
σ
‘
and
σ
n
σ
m
.
Solution:
σ
‘
(
x,y
) = (4

x,y
),
σ
m
(
x,y
) = (
x,
6

y
) and
σ
n
(
x,y
) = (
x,
10

y
). It follows
that
σ
m
σ
‘
(
x,y
) =
σ
m
(4

x,y
) = (4

x,
6

y
) and that
σ
n
σ
m
(
x,y
) =
σ
n
(
x,
6

y
) =
(
x,
10

(6

y
)) = (
x,y
+ 4).
2. Let
A
= (0
,
0),
B
= (5
,
0),
C
= (0
,
10),
D
= (4
,
2),
E
= (1
,

2),
F
= (12
,

4). Find lines
‘,‘
0
,‘
00
so that the products of the reﬂections in
‘,‘
0
,‘
00
map
4
ABC
to
4
DEF
.
Solution:
This problem was stated incorrectly!!!!
you can ﬁnd two lines
‘
and
‘
0
so
that the reﬂection in these two lines sends
4
ABC
to
4
DEF
(it cannot be done with three).
This is the problem we solve below.
First we ﬁnd a reﬂection
σ
‘
which maps
A
to
D
. The line through (0
,
0) and (4
,
2) has
slope
1
2
so
‘
must have slope

2. The line
‘
must also contain the midpoint of the line segment