Alice in Wonderland
By Lizard and Breck
Chapter 10
Lesson #1
Tangents to Circles
…when suddenly a White Rabbit with pink eyes
ran close by her. There was nothing so very
remarkable in that; nor did Alice think it so very
much out of the way to hear the Rabbit say to
itself, “Oh, dear! Oh, dear! I shall be too late!”
But when the rabbit actually took a watch out of
its waistcoat pocket…..
10.1
…Alice is so very surprised by
this occurrence that she
decides to identify every part of
the pocket-watch, which,
coincidently, is a circle. Can
you help her? (She is not a
terribly clever girl).
AB is a ________.
CB is a ________.
A is the _______.
BD is a ________.
EF is a _________.
GH is a ________.
I is the ________.
10.1
Alice also notes that a line
is tangent to a pocket
watch (or circle) iff it is
perpendicular to the radius
drawn to the point of
tangency.
What theorems are these?
Lesson #2
Arcs and Chords
…before she found herself falling down what
seemed to be a very deep well…Then she looked
at the sides of the well, and noticed that they were
filled with cupboards and bookshelves. She took
down a jar from one of the shelves as she passed.
It was labeled “ORANGE MARMALADE,” but
to her great disappointment it was empty.
10.2
Given that falling down a large well gets quite dull
after a while, Alice started to play with the empty
marmalade jar and some string. With the string
she formed one central angle on the jar’s mouth,
which in turn created a major arc and a minor arc.
But Alice, being not terribly clever, forgot which
arc is which. Kindly point out the major and
minor arcs for her?
10.2
While falling Alice also picks up a slip of paper that
reads:
“Theorem 10.4: In the same circle or in congruent circles
two minor arcs are congruent if and only if their
corresponding chords are congruent.
Theorem 10.5: If a diameter of a circle is perpendicular to
a chord, then the diameter bisects the chord and its arc.
Theorem 10.6: If one chord is a perpendicular bisector of
another chord, then the first chord is a diameter.”
Theorem 10.7: In the same or congruent circles, two
chords are congruent iff they are equidistant from the
center.
10.2
When Alice
found another
marmalade jar
in the well,
she created a
diagram for
one of the theorems she had just learned…but then
forgot which one. Can you help her?
Lesson #3
Inscribed Angles
She stretched herself up on tiptoe, and peeped
over the edge of the mushroom, and her eyes
immediately met those of a large blue caterpillar,
that was sitting on the top, with its arms folded,
quietly smoking a long hookah, and taking not the
smallest notice of her or of anything else.
10.3
Alice simply could not stop staring at the awesome
blue smoke rings that the caterpillar was creating.
Yet, with all of
that geometry
in her head at
the time, she
had to notice
the inscribed
angles inside
the rings.
10.3
Alice also noticed
something unusual about
the angle inscribed in the
smoke ring. Its measure
was half the measure of
its intercepted arc! So, if
AB is the diameter,
what is the measure of
angle ACB?
10.3
While she left- the caterpillar whispered in her ear
the remaining theorems…..
Theorem 10.9: If two inscribed angles of a circle
intercept the same arc, then the angles are
congruent.
Theorem 10.10: If a right triangle is inscribed in a
circle, then the hypotenuse is a diameter of the
circle.
Theorem 10.11: A quadrilateral can be inscribed in
a circle iff its opposite angles are supplementary.
Lesson #4
Other Angle Relationships in Circles
The March Hare took the watch and looked at it
gloomily. Then he dipped it into his cup of tea, and
looked at it again.
10.4
While in the company of the Mad Tea Party, Alice heard many odd
things, but none so odd as the next few fascinating theorems:
Theorem 10.12: If a tangent and a chord intersect a point on a
circle, then the measure of each angle formed is one half the
measure of its intercepted arc.
Theorem 10.13: If two chords intersect in the interior of a circle,
then the measure of each angle is one half the sum of the measures
of the arcs intercepted by the angle and its vertical angle.
Theorem 10.14: If a tangent and a secant, two tangents, or two
secants intersect in the exterior of a circle, then the measure of the
angle formed is one half the difference of the measures of the
intercepted arcs.
10.4
While Alice was observing this strange tradition,
she noticed that a jam stain on the table formed a
tangent line to a circular teacup, and a chopstick
left atop the cup formed a secant. Given that the
two intersected on the outside of the circle, what is
the measure of the angle formed by the two?
10.4
Lesson #5
Segment Lengths in Circles
Then the Queen left off, quite out of breath, and
said to Alice, “Have you seen the Mock Turtle
yet?” “No,” said Alice. “I don’t even know what a
Mock Turtle is.” “It’s the thing Mock Turtle Soup
is made from,” said the Queen.
10.5
Alice, beginning to grow tired of geometry
connections in her adventure, decides to entertain
a new guest, the Mock Turtle. Little does she
know, the Mock Turtle has a perfect example of
theorem on his back, and he soon explains three
new theorems to Alice.
10.5
Theorem 10.15: If two chords intersect in the interior of
a circle, then the product of the lengths of the segments
of one chord is equal to the product of the lengths of the
segments of the other chord.
Theorem 10.16: If two secant segments share the same
endpoint outside a circle, then the product of the length
of one secant segment and the length of its external
segment equals the corresponding product of the other.
Theorem 10.17: If a secant segment and a tangent
segment share an endpoint outside a circle, then the
product of the length of one secant segment and the
length of its external segment equals the square of the
length of the tangent segment.
10.5
As Alice is very forgetful…..which theorem does the
Mock Turtle
B
represent on
his back, and
A
what does this
theorem tell
us?
D
C
E
Lesson #6
Equations of Circles
Alice had never been in a court of justice before,
but she was quite pleased to find that she knew
the name of nearly everything there. “That’s the
judge,” she said to herself, “because of his wig.”
10.6
10.6
So, if one tart has a center located at the origin,
with a radius of 2, what is the equation of the
circle?
What about a tart with a
center at (-2, 3), and a
radius of 3?
Chapter 11 with Alice
Areas of Polygons and Circles
BRECK RADULOVIC
AND
ELIZABETH FRYAR
11.1 Angle Measures in Polygons
“You out to be ashamed of yourself,” said Alice,
“a great girl like you,” (she might as well say this)
“to go on crying like this!”
11.1 Angle Measures in Polygons
 Poor Alice, she had grown very tall after eating her
cake. While little Alice was mourning her new height,
White Rabbit came hurrying back into the hall,
saying, “Won’t the Duchess be savage if I keep her
waiting?” Of course Alice wondered what the
Duchess was waiting on. She was much too afraid to
ask, however, and the little rabbit ran away before
she could find out. As she was crying, she noticed
that White Rabbit had dropped a piece of paper
which was addressed to the Duchess…
11.1
 To the Duchess
I know you have asked for some theorems on the Angle Measures in
Polygons. I have searched all over Wonderland and found these two
theoremsTheorem 11.1- Polygon Interior Angles Theorem: the sum of the
measures of the interior angles of a convex n-gon is (n-2) x 180.
 The corollary to this theorem is: the measure of each interior angle
of a n-gon is 1/n x (n-2) x 180; or [(n-2) x 180]/n.
 Theorem 11.2- Polygon Exterior Angles Theorem: the sum of the
measures of the exterior angles of a convex polygon, one angle at each
vertex, is 360.
 The corollary to this theorem is: the measure of each exterior angle
of a regular n-gon is 1/n x 360; or 360/n.
Your Loving Servant,
White Rabbit

11.1
 After learning these theorems, Alice realized she was
sitting in a pool of her own tears. She was so
devastated that she had made a river simply by
crying, she cried some more. When Alice finally
calmed down, she realized her tears formed the sides
of a regular 27-gon. She wondered what the sum of
all the interior angels were and how many degrees
each angle had. Alice was determined to use both
theorems, so she also wondered how many degrees
each exterior angle had. Unfortunately, Alice was
feeling awfully stupid, and needed some help. Can
you do so?
11.1
 Sum of Interior Angles- 4500
 Interior Angle Measure- 166 2/3
 Exterior Angel Measure- 13 1/3
 Alice was very grateful for your help. So grateful, she
began to shrink… and shrink… and
shrink…
11.2 Areas of Regular Polygons
“I know something interesting is sure to happen,”
she said to herself, “whenever I eat or drink
anything: so I’ll just see what this bottle does. I do
hope it’ll make me grow large again, for really I’m
quite tired of being such a tiny little thing.”
11.2 Areas of Regular Polygons
 Alice had grown so much she had overtaken W.
Rabbit’s entire house. W. Rabbit was understandably
very frightened and gathered his neighbors to try and
get Alice out. They tried and tried, but Alice’s arm
was still hanging through Rabbit’s bedroom window.
The animals finally decided that they must send Bill
the Lizard into the house to get rid of Alice. Alice
gave Bill a great kick and sent him flying into the
hedge. Bill was so awfully surprised, he blurted out
the first thing he could remember, theorems about
the area of regular polygons…
11.2
 The first theorem Bill stated was Theorem 11.3- Area
of an Equilateral Triangle: the area of an equilateral
triangle is one fourth the square of the length of the
side times √3. A= ¼ √3 s2.
 The second theorem was Theorem 11.4- Area of a
Regular Polygon: the area of a regular n-gon with
side length s is half the product of the apothem a
and the perimeter P, so A= ½ aP, or A = ½ a x ns.
11.2
 Alice was indeed dreadfully tired of being so very
large when she noticed another cake. She noted that
the cake was a nonagon with side length 3 cm. She
wanted to know the area of her cake using the second
theorem. She is not yet confident with the theorem
and would like your help…
11.2
 The area is 55.64 cm.
 Alice is once again quite grateful for your help in
finding the area of her cake. After shrinking-thanks
to the help of her cake- she left W. Rabbit’s house
and continued through the beautiful garden where
she saw a certain caterpillar…
Perimeters and Areas of Similar Figures 11.3
“But I don’t want to go among mad people,” Alice
complained.
“Oh, you can’t help that,” said the Cat: “we’re all mad
here. I’m mad. You’re mad.”
Perimeters and Areas of Similar Figures 11.3
 Alice wandered about until she reached a house; the
Duchess’ house. The air was full of pepper and Alice
was having much difficulty not sneezing. But the
strangest thing of all was the Duchess’ baby, because
it was not a baby, it was a pig. Alice ran off from the
house and once again met the Cheshire Cat. The Cat
informed her that she shall play croquet with the
Queen, and therefore needed to know a theorem…
11.3
 “You must know of Theorem 11.5- Areas of Similar
Polygons,” the Cat said. Alice did not, however, know
of this theorem, and so the Cat told her:
“If two polygons are similar with the lengths of
corresponding sides in the ratio of a:b, then the
ration of their areas is a2:b2.”
11.3
 As Alice was talking to the Cat, he proceeded to
slowly disappear, but always in the shape of a
trapezoid. The bases of his head the first time were 3
and 2 inches respectively. The height was 4. The
Cat’s head then shrunk and had an area of 6 2/3.
What were the dimensions of the second trapezoid?
11.3
 The original dimensions were
 Height – 3.18 in
 Base 1- 1.59 in
 Base 2- 2.39 in
 As the Cheshire Cat finally disappeared, Alice was on
her way. She came to a fork in the road, one way led
to the March Hare, and the other to the Mad Hatter.
Alice continued along to the house of the March
Hare.
Circumference and Arc Length 11.4
“First came ten soldiers carrying clubs: these
were all shaped like the three gardeners, oblong
and flat, with their hands and feet at the corners;
next the ten courtiers: these were ornamented all
over with diamonds, and walked two and two, as
the soldiers did.”
Circumference and Arc Length 11.4
 Alice came along to a group of gardeners who were
painting white roses red. As Alice was watching, the
Queen arrived with all her soldiers and children and
the King. The Queen suddenly became very unhappy
with Alice. She announced that if Alice did not know
theorems 11.6 and its corollary, it would be “Off with
her head!”…
11.4
 Luckily, Alice did know the theorems.
 Theorem 11.6- Circumference of a Circle: the circumference C
of a circle is C = 2πr where r is the radius of the circle.

The corollary, Arc Length Corollary: In a circle, the ratio of the
length of a given arc to the circumference is equal to the ratio of
the measure of the arc to 360 degrees.
11.4
 The Queen was impressed with Alice’s knowledge
and so invited her to play croquet. Alice was forced
to play croquet with flamingos for mallets and
hedgehogs for balls. The Queen was always the first
to go and the last to go and all the ones in between.
The Queen told Alice that if she answered the
following conundrum correctly, the girl would be
allowed to try and hit the hedgehog. “If I swing my
flamingo in a circle with a diameter of 9y and will be
able hit the hedgehog for 40 degrees which will be
1.25 x, what will x and y be?”
11.4
 x= 1.25
 y = 3.5
 Alice answered correctly to the Queen, and should
have been allowed to play, but her hedgehog crawled
away. She was disappointed, but the Cheshire Cat
called her away to meet the Mock Turtle and the
Gryphon.
Areas of the Circles and Sectors 11.5
“Alice said nothing: she had sat down with her
face in her hands, wondering if anything would ever
happen in a natural way again.”
Areas of Circles and Sectors 11.5
 The Mock Turtle and the Gryphon told Alice that
they would perform a song. Alice had not ever heard
this “Lobster-Quadrille” before and thought it very
interesting, but was very glad when it was finally
over. The Mock Turtle said that the Gryphon had left
out a verse of the Quadrille, one that stated the
theorems about Areas of Circles and Sectors…
11.5
 “Well… there was a theorem 11.7 which told of the
Area of a Circle….
The area of a circle is π times the square…
of the raaaadius…
And OH, but there is more… the theorem 11.8 which
relates… the Area of a Sector… The ratio of the area
A of a sector… of a circle to the areaaaaa… of the
circle… is equal to the ratiooooo… of the measure of
the intercepteddddd arc… to 300 and 60 degrees!”
11.5
 Alice was rather confused by this song’s rhythm but
the Gryphon was too busy lauding his friend’s song
to answer Alice’s question. The Mock Turtle sang
another song to test Alice’s knowledge of sector
areas. The song was so horrifically awful and off-tune
that Alice blocked it from her memory. She did,
however, remember the problem that was posed: If I
have a circle with diameter 37.5 fathoms and sector
that measures in degrees 87.62% of the radius
squared minus the diameter, what is the area of my
sector?
11.5
 830 fathoms2
 Once Alice finally got this question right, the
Gryphon took her back along the way to the Queen
who was in a great fit. Someone, you see, had stolen
the Queen’s tarts!
Geometric Probability 11.6
“At this time the whole pack rose up into the air
and came flying down upon her; she gave a little
scream, half of fright and half of anger, and tried to
beat them off…”
Geometric Probability 11.6
 Alice was called to court to testify in the case of the
missing tarts that once belonged the Queen of
Hearts. Alice arrived later, once the proceedings had
begun. The knave was on the witness stand and the
Queen was accusing him of stealing her tarts! She
said that he must be guilty if he could not tell her
about Geometric Probability. The poor knave,
though guilty, was so in shock he could not, but he
rather liked his head. Alice liked his head too, so she
decided to whisper the answers to him, for the
Gryphon had relayed the information to her on the
way to the court…
11.6
 Alice told the little Knave
 Probability and Length: Let AB be a segment that contains the
segment CD. If a point K on AM is chosen at random, then the
probability that it is on CD is as follows:


P(Point K is on CD) = Length of CD/ Length of AB
Probability and Area: Let J be a region that contains region M.
If a point K in J is chosen at random, then the probability that
it is in region M is as follows:

P(Point K is in region M) = Area of M/ Area of J
11.6
Hint: You don’t actually need
Alice’s size!
 The Queen soon found out that Alice had helped the
Knave and commanded her cards to go after the girl.
Alice felt herself start to grow. She knew she would soon
be quite a bit taller than she was now, and wanted to
know the probability of landing on the jury who were
seated in the jury box. If Alice is currently three inches
tall by 1 inch wide and .5 in thick and in the shape of a
rectangular prism standing in a cylinder shaped room
with radius 17 in and height 16 in, with a jury box in the
shape of a sphere with a radius of 7 in, what is the
probability she will land on the jury, and poor Bill the
Lizard if she increases her size by a factor of 4?
11.6
 The Probability she lands on the jury is 9.89%
 Alice did indeed land on the jury, and while Bill the
Lizard was screaming about his broken leg, she
managed to escape from Wonderland. When she
arrived back home she awoke under a tree in her
sister’s lap. When Alice told her sister of her
marvelous adventure in Wonderland, her sister
laughed and told her that she had the craziest
dreams…