Alice in WonderlandChapter 10

By Liza

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Lesson #1Tangents 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 #2Arcs 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?

10.2While 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 #3Inscribed 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.

10.3Alice 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?

10.3While 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 #4Other 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 #5Segment 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.

10.5Theorem 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.

10.5As Alice is very forgetful…..which theorem does the Mock Turtle represent on his back, and what does this theorem tell us?

A

B

C

D

E

Lesson #6Equations 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?

BRECK RADULOVICAND

ELIZABETH FRYAR

Chapter 11 with AliceAreas of Polygons and

Circles

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 theorems- Theorem 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- 4500Interior 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= 1.25y = 3.5Alice 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

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?

Hint: You don’t actually need Alice’s size!

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…