MATH 680 Problem Set 5 due July 18 th (7 points) If you work with anyone else on these, please indicate this on your problem set. Cryptography problems A B C D E F G H I J K L M N O P Q R S T U V W X...

MATH 680 Problem Set 5 due July 18^th



(7 points)

If you work with anyone else on these, please indicate this on your problem set.

Cryptography problems

A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V	W	X	Y	Z
0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25

Decrypt the Ciphertext [4 0 30 16 38 28 22 14 16 22 22] that was encrypted using the equation,
Ciphertext = Plaintext * 2
Represent the Plaintext in letters. Show your work. (1 point)
Encrypt the Plaintext, “CHERRYBLOSSOM” using the Vigenere cipher and the word “DC.”
Represent the Ciphertext using letters. Show your work. (1 point)

Geometry problems


“17” Problem

The horizontal and vertical distances between pegs on the square-array geoboard shown here are equal. If the outlined figures have a total area of 17 units², how far apart are consecutive horizontal pegs?
(1 point)



The lunes and the triangle

In the diagram below, DFE is a right triangle, where angle DFE is the right angle. Points A, B, and C are the midpoints of line segments DF, FE, and ED respectively. Three circles are drawn with centers at A, B, and C, with diameters DF, FE, and ED respectively.
Given that you know the Pythagorean theorem holds between sides of a right triangle, prove that the sum of the areas of the semicircles on the legs of a right triangle is equal to the area of the semicircle on the hypotenuse. Refer to the diagram below in your proof.
(2 points)








Traveling Triangle

The diagram below shows a square withsides of length 2 inches and an equilateraltriangle
ABC
with sides of length 1 inch sittinginside it. The vertex
B
is at one corner of thesquare and the side
BC
lies along the square’sbottom edge. The triangle is rotated in aclockwise direction about its corners
C,
A,
B
inturn and rolls without slipping inside of thesquare. If the triangle is rotated until the cornersA,
B, and
C
have returned to their
originalpositions, what is the

total distance

travelled bythe point
A?

Hint #1: You may want to create a visual model for yourself with a triangle and a square to see what happens.


Hint #2: What geometric shape does the trajectory of point A resemble?


(2 points)