Assignment #1
Solutions
Math 300
1. (a) Omitted.
(b) 1.3.1. In the triangle ABD, we have that AB = 2, BD = 1, AD = 5 and
AG = x. Since G is a point on AB that splits it into a golden ratio, we have
AB
1+ 5
=
AG
2
2
1+ 5
=
x
2
4
= x
1+ 5
4(1 5)
= x
(1 + 5
Assignment #2
Solutions
Math 300
1. (a) In rectangle ABCD, let E be a point on AB and F be a point on CD so that
EF is perpendicular to both AB and CD. Let a = AD, b = DF and c = F C.
Then
a(b + c) = area of ABCD
= area of AEF D + area of
= ab + ac
EBCF
(
Assignment #4
Solutions
Math 300
1. Let A , B and C be the midpoints of BC, AC and AB respectively. Let X be the
centroid of ABC.
(a) Triangles BXA and CXA have the same height from vertex X and since
BA CA they have the same base. It follows that area(BX
Assignment #3
Solutions
Math 300
1. (a) Suppose ABC has B C. Let AD be perpendicular to BC and suppose
=
AD intersects BC at X. Then AX AX, ABX ACX and AXB
=
=
=
AXC (both are right angles). By AAS, AXB AXC. It follows that
=
AB AC and so ABC is isoscele
Assignment #6
Solutions
Math 300
1. Let R be a point on AB outside of ABC. Without loss of generality, assume A is
between R and B. Draw a line through R that intersects AC at P and BC at Q. Con
struct a line parallel to AC through B. Let S be the point o
Assignment #7
Solutions
Math 300
1. Note that OC OG since both are radii of the circle centered at O with radius
=
OB. Also BC BE since both are radii of the circle centered at B with radius
=
BO. It follows that OC BC and OG BE. Since CG and CE are radii
Assignment #9
Solutions
Math 300
1. (a) Suppose T = rm r is a translation with translation vector v. Note that the line
and m are parallel. Then
T (r rm ) =
=
=
=
=
(rm r ) (r rm )
rm (r r ) rm
rm id rm
rm rm
id
Similarly (r rm ) T = id. So, r rm is the i
Assignment #8
Solutions
1. (a)
Math 300
i. Suppose f (x) = y. Then f 1 (y) = x.
(f 1 f )(x) = f 1 (f (x)
= f 1 (y)
= x
So, f 1 f = id.
(f f 1 )(y) = f (f 1 (y)
= f (x)
= y
So f f 1 = id
ii. Suppose f has two inverses, g and h. Then f g = id = f h and h f
Test #1
Solutions
Math 300
In-Class Test
1. (a) An axiomatic system is consistent if no two statements (these could be two axioms,
two theorems or an axiom and a theorem) contradict each other.
(b) A cevian of a triangle is a segment from a vertex to a po