MAT 336 Homework 4

Notes

• In order to get full credit in the open ended questions, you need to spend some time reflecting upon and writing your answers. In most of these problems, the probability that a one-sentence answer will get full credit is close to 0 (and so will be what you learn from the exercise)
• Make sure you show all your work on the problems that require it so. Otherwise, even if you give a correct answer, if you do not explain how you obtained it, you'll get very little or no credit.
• It would be great if you discussed ideas with your classmates. The write-up, however, must be done individually.
• Recall that the slides of the lectures can be found here.
• Follow the AI policy.

Problems

Suggestion to better understand Euclid : Play Euclidea

1. Consider a right triangle T whose legs have length 2 and 9. Divide T into polygonal pieces from which a square can be assembled. What is the side length of this square? Hint: First divide the triangle into pieces that can be reassembled into a rectangle. You can use cm or any other unit of your choice. "Make" your solution on paper or in GeoGebra. Explain how you obtained the solution and provide photos or screenshots of the pieces you made. Include measurements marked on your drawing. (If you complete this successfully, you will have produced a quadrature of a right triangle!)
2. Using Euclid’s idea from Proposition IX.20, choose three primes (include 2 and 41) and create a new number by multiplying them and adding 1. Explain why Euclid’s reasoning shows that the new number has a prime factor not in the original list. Write a short paragraph explaining how Euclid’s approach differs from the modern statement “there are infinitely many primes.”
3. Given A, B, C on a line with B between A and C, and BC = 1, construct a segment of length √AB. Use straightedge and compass (or GeoGebra). Show and explain your steps. What proposition in Euclid's Elements does this resemble? (Note: You can use AI for this last question, but make sure you check your answer. Here is a copy of Euclid's Elements)
4. Summarize Euclid’s proof of the Pythagorean Theorem (as discussed in class), highlighting the main steps and constructions. Optionally include a sketch, or refer to the GeoGebra activity we used. What earlier propositions does the proof depend on? (Here is a copy of Euclid's Elements)
5. Extra credit: In Proposition 14 of Book 2 of Euclid’s Elements, a construction for squaring a “rectilinear figure A” is given. Perform the construction in this GeoGebra activity starting with a rectangle BCDE (instead of a general rectilinear figure A). That is, in the fourth sentence of the proof (“Then, if BE equals ED, then that which was proposed is done, for a square BD has been constructed equal to the rectilinear figure A.”). Construct a square of side length ED, and check in GeoGebra using the Area tool that the areas of rectangle BCDE and the square of side ED are equal. Send a screenshot for credit.

Sample Quiz 4

1. Consider a right triangle T whose legs have length 2 and 9. Divide T into polygonal pieces from which a square can be assembled. What is the side length of this square? Hint: First divide the triangle into pieces that can be reassembled into a rectangle. You can use cm or any other unit of your choice. "Make" your solution on paper or in GeoGebra. Explain how you obtained the solution and provide photos or screenshots of the pieces you made. Include measurements marked on your drawing. (If you complete this successfully, you will have produced a quadrature of a right triangle!)
2. Restate Euclid’s Proposition I.1 in your own words. Explain why the construction produces an equilateral triangle, citing which postulates or common notions justify each step. Justify your steps by the appropriate Common Notions or Postulates (This will be given in the quiz, now you can find them in the Lecture Slides.) Use complete sentences.
3. Using Euclid’s idea from Proposition IX.20, choose three primes (include 2 and 41) and create a new number by multiplying them and adding 1. Explain why Euclid’s reasoning shows that the new number has a prime factor not in the original list. Write a short paragraph explaining how Euclid’s approach differs from the modern statement “there are infinitely many primes.”