Math 124
Exam 1
Solutions
1. Define each underlined word:
a.
α:R2→R2 is a transformation.
α is a transformation if and only if α is 1-1 and onto.
b. α:R2→R2 is a dilatation.
α is a dilatation if and only if it is a collineation and, for any line m, α(m)
and m are parallel.
2. Complete each of the following.
a. If P is a point and α is the rotation about P of 350o, the order of α is 36.
b. The order of a non-identity translation is infinite.
c. Point P is fixed by halfturn σQ if and only if P=Q.
d. Line m is fixed by halfturn σQ if and only if Q is on m.
↔
e. Line m is fixed by translation τPQ if and only if line m and line PQ are parallel.
f. Line m is fixed by reflection σn if and only if either m=n or m and n are
perpendicular.
g. Line m is fixed pointwise by reflection σn if and only if m=n.
3. Consider the points P, Q, and R is the picture below.
.C
.Q
.B
.D
P
.
.R
.A
a. Label the point A so that σA = σP σQ σR .
b. Label the point B = τPQ (R).
c. Label the point C = σ ↔ (R).
PQ
d. Label the point D so that τPQ = σQ σD
4. Determine the image of the line 3X+8Y = 5 under the halfturn σQ where Q = (-4,5).
The halfturn σQ is given by:
x' = -x-8
y' = -y+10
Let (x,y) be an arbitrary point in the plane.
(x,y) is on 3X+8Y = 5 iff
3x+8y=5 iff
3(-x'-8) + 8(-y'+10) = 5 iff
-3x' –8y' = 24 – 80 + 5 iff
-3x' –8y' = -51 iff
3x' +8y' = 51 iff
(x',y') is on 3X+8Y = 51
This tells us that the image of the line 3X+8Y = 5 under the halfturn σQ is
3X+8Y = 51.
5. Suppose σP(12,-3) = (10,11). Determine the point P.
P = the midpoint of the line segment from (12,-3) to (10,11) = (11,4).
6. Suppose τPQ (1,8) = (3,-3) and Q=(8,-2). Determine the point P.
τPQ (1,8) = (3,-3) tells us that τPQ is given by
x' = x+2
y' = y-11.
Then, since Q=(8,-2), we know that P = (8-2,-2+11) = (6,9)
7. Each of the following transformations is either a translation or a halfturn. Decide
which and justify your answer. Assume A, B, C, and D are distinct points.
a. τAB τAC σD
The composition of a halfturn and a translation is a halfturn. Hence, τACσD
is a halfturn. Then the composition of translation τAB and halfturn τAC σD is
a halfturn. Thus, τAB τAC σD is a halfturn. (There are other approaches.)
b. σA σB σC σD
The composition of two halfturns is a translation. Hence, σAσB and σ CσD
are each translations. The composition of these two translations is a
translation. Thus, σAσBσCσD is a translation. (There are other approaches.)
8. Give an example of each of the following:
a. A transformation that is not an isometry.
Examples:
i. The transformation that doubles the distance, in the same direction, from
a given point.
ii. α(x,y) = (x,y3)
(Many other examples are possible.)
b. A collineation that is not a dilatation.
Examples:
i. A rotation about a given point of 11o.
ii. A reflection about a given line.
(Many other examples are possible.)
c. A transformation of order five.
Example:
A rotation about a given point of 72o.
9. Prove: A halfturn is an isometry.
Proof: Let P=(a,b) be an arbitrary point and consider the halfturn σP . We
must show that σP is an isometry.
Let Q=(c,d) and R=(e,f) be arbitrary points. To show that σ P is an
isometry, we must show that QR = σ P(Q)σP(R). Equations for σP are
given by:
x' = -x + 2a
y' = -y + 2b
We show that QR = σP (Q)σP (R) as follows:
σP (Q)σP (R) =
(-c+2a,-d+2b) (-e+2a,-f+2b) =
(c − e) + (d − f )
2
2
=
QR
This establishes that σP is an isometry, as desired.
____
10. Suppose Q is the midpoint of segment PR . Prove: σQ σP = τP R.
⎛a +c b+d⎞
,
Proof: Let P=(a,b) and R=(c,d). Then Q = ⎜
⎟ . For any point
⎝ 2
2 ⎠
(x,y),
σQ σP (x,y) =
σQ (-x+2a,-y+2b) =
(x-2a+a+c,y-2b+b+d) =
(x+c-a,y+d-b) =
τP R (x,y)
This establishes that σQ σP = τP R , as desired.