Geometry

1. Geometry Chapter 10 sec 1

2. Who was Euclid? • 300 BC, also known at Euclid of Alexandria. • He was a Greek mathematician, referred to as the “Father of Geometry”. • Elements written by Euclid and is an introduction to elementary mathematics. It included number theory and geometry.

3. Points, lines, and Circles. • Euclid begins the topic of geometry with intuitive descriptions of three undefined terms. • 1. Point – “that which has no part” • 2. Line - “length but no breadth” • 3. Plane – “ length and breadth only”

4. Terminology • A • The picture above is a line. • A • Ray- is a half line with its end point included.

5. Line segment – A piece of a line joined two points and including the points. • A B • According to Euclid, lines either intersect or are parallel.

6. Angles • Angle – two rays having a common endpoint. C A B • Ray AB is called the initial side. • Ray BC is called the terminal side. • Point B is called the vertex.

7. Measurement of angles • Degree – measurement of the angle. • If you start from the initial side and rotate one complete revolution about the vertex so that the terminal side coincides with the initial side, you will have formed 360.

8. Acute angle • Acute angle – is an angle that has a measurement between 0 and 90. 37

9. Obtuse angle • Obtuse angle – is an angle that has a measurement between 90 and 180. 157

10. Right angle • Right angle – has a measurement of 90.

11. Complementary and Supplementary angle • Complementary angle – if the sum of their measurements is 90. • Supplementary angle – if the sum of their measurements is 180.

12. Transversal • If we intersect a pair of parallel lines with a third line. We have then 8 angles to consider.

13. 1 2 3 4 5 6 7 8

15. History of Circles • The circle has been known since before the beginning of recorded history. It is the basis for the wheel which, with related inventions such as gears, makes much of modern civilization possible. In mathematics, the study of the circle has helped inspire the development of geometry and calculus. • Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.

16. Some highlights in the history of the circle are: • 1700 BC – TheRhindpapyrus gives a method to find the area of a circular field. The result corresponds to 256/81 (3.16049...) as an approximate value of π. • 300 BC – Book 3 of Euclid's Elements deals with the properties of circles. • 1880 – Lindemannproves that π is transcendental, effectively settling the millennia-old problem of squaring the circle. • wikipidea

17. Circles • Circle – is the set of all points lying on a plane that are located at a fixed distance, called the radius, from a given point called the center. • Radius – it is the distance between the center and any point on the circle. • Diameter – is a line segment passing through the center with both endpoints lying on the circle.

18. Circumference is the distance around the circle. • See next slide

19. Circumference radius Diameter

20. Pi • The constant is also known as Archimedes Constant, although this name is rather uncommon in modern, western, English-speaking contexts. Many formulae from mathematics, science, and engineering involve π, which is one of the most important mathematical and physical constants.

21. History of Pi • The earliest evidenced conscious use of an accurate approximation for the length of a circumference with respect to its radius is of 3+1/7 in the designs of the Old Kingdom pyramids in Egypt. The Great Pyramid at Giza, constructed c.2550-2500 BC, was built with a perimeter of 1760 cubits and a height of 280 cubits; the ratio 1760/280 ≈ 2π.

22. Egyptologists such as Professors Flinders Petrie and I.E.S Edwardshave shown that these circular proportions were deliberately chosen for symbolic reasons by the Old Kingdom scribes and architects.

23. The same apotropaic(Intended to ward off evil) proportions were used earlier at the Pyramid of Meidum c.2600 BC. This application is archaeologically evidenced, whereas textual evidence does not survive from this early period.

24. The early history of π from textual sources roughly parallels the development of mathematics as a whole. Some authors divide progress into three periods: the ancient period during which π was studied geometrically, the classical era following the development of calculus in Europe around the 17th century, and the age of digital computers.

25. Antiquity • That the ratio of the circumference to the diameter of a circle is the same for all circles, and that it is slightly more than 3, was known to Ancient Egyptian, Babylonian, Indian and Greek geometers.

26. The earliest known textually evidenced approximations date from around 1900 BC; they are 25/8 (Babylonia) and 256/81 (Egypt), both within 1% of the true value. The Indian text ShatapathaBrahmana gives π as 339/108 ≈ 3.139.

27. Story about the circumference of the earth. • How many of you know the circumference of the earth? • How many of you know how to calculate the circumference of the earth?

28. Before we answer these questions, here is a small history of Pi.

29. Eratosthenes • Eratostheneswas a Greek astronomer in ancient times. Around 240 B.C. he made the first good measurement of the size of the Earth.

30. Who was he? • Eratosthenes lived in the city of Alexandria. Alexandria is in northern Egypt. It is by the Nile River and the Mediterranean Sea. There was a tall tower in Alexandria. Eratosthenes measured the length of the tower's shadow on the Summer Solstice. He used that information plus some geometry to figure out the angle between the Sun and straight up.

31. Eratosthenes used the lengths of shadows to figure out how high in the sky the Sun was in a certain place on a certain day. He knew of another place where there was no shadow at all on the same day. That meant the Sun was straight overhead. He found out the distance between the two places, then used some geometry to figure out the rest.

32. There was a town in southern Egypt called Syene. There was a well in Syene. On the Summer Solstice, the Sun shone straight down the well to the very bottom. That meant the Sun must be straight overhead.

33. Eratosthenes had someone measure the distance between Alexandria and Syene. He used that distance, what he knew about the Sun's angles, and a bit of geometry to figure out the size of the Earth.

34. We aren't quite sure what answer Eratosthenes came up with, though. The distance between Alexandria and Syene was measured in stadia. The stadion was a distance unit that was often used in ancient times. However, not everybody used a stadion of the same length.

35. If Eratosthenes used one length for the stadion, his answer was really, really good. The Earth is about 40 thousand kilometers (about 24,860 miles) around. The measurement that Eratosthenes made might have been within about 1% of this. That would be amazing! However, he might have used a different length stadion. If that is true, his answer was off by about 16%. That is still pretty good!