PROPOSITION 48. THEOREM

PROPOSITION 

48

.  THEOREM

If the square drawn on one side of a triangle is equal to the squares drawn on the other two sides, then the angle contained by those two sides is a right angle.

It is this proposition that informs us that if the sides of a triangle are 3-4-5 — so that the squares on them are 9-16-25 — then the triangle is right-angled. Whole-number sides such as those are called 

Pythagorean triples

.

For the proof, see Problem 4.

Pythagoras and the Pythagoreans

Pythagoras is remembered as the first to take mathematics seriously in relation to the world order. He taught that geometry and numbers should be studied with reverence, because we are entering into knowledge of the Divine. Mathematics is therefore more than just intellectual stimulation, and its relationship to the Universe is more than just coincidence.

Pythagoras was born on the Greek island of Samos. He left to found what we might call a religious sect in the city of Crotona in southern Italy, which was then part of greater Greece. His devotees were called Pythagoreans ("Pi-thag-o-REE-ans"), and many of them, both men and women, lived communally. They had their rituals and their dietary laws, and they made important contributions to the medicine and astronomy of their time. They were among the first to teach that the earth is round and that it revolves about the sun.

They also taught the continuity and reincarnation of life; and that through philosophy (literally, love of wisdom) there is purification and thus escape from the cycle of births. Hence they practiced equality between one another, and they showed compassion to all creatures, because we are all forms of One.

The Pythagoreans were the first to systematically investigate both arithmetic and geometry. Not only did they discover many theorems, but they gave an ethical and spiritual significance to each of the figures they drew. As a secret society, it must be said that they were extremely successful — because not a word has survived! In this regard, Pythagoras is credited with discovering incommensurable magnitudes. (See Topic 10 of The Evolution of the Real Numbers.) They considered their doctrine of incommensurables their most esoteric teaching — one of their members was treated as dead for having even spoken of it to an outsider. The significance they gave to it has never been revealed.

As a scientific researcher, Pythagoras discovered how musical harmonies depend on ratios of whole numbers. He found that we

hear the interval called an octave when the length of a vibrating string is halved, that is to say, when the length of the plucked string is to the whole string in the ratio 1:2 (Fig 1). We hear the interval called a perfect fifth (G above C) when two thirds of a string is plucked (Fig. 2), that is, when the stopped string is to the whole string in the ratio 2:3. And we hear a perfect fourth (F above C) when three fourths of the string is sounded (Fig. 3); and so on, for each interval in the musical scale.

In fact, Pythagoras taught that all things are known through number (not symbols for numbers). Or rather, "All things are assimilated to number." This means that just as one population becomes assimilated by another — they become just like those others — so all things are like numbers. In that likeness, all things meet and are intelligible.

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THE PYTHAGOREAN THEOREM: Book I. Propositions 47 and 48

P l a n e   G e o m e t r y

An Adventure in Language and Logic

based on


THE PYTHAGOREAN THEOREM

Book I.  Propositions 47 and 48

Proposition 47

P

YTHAGORAS was a teacher and philosopher who lived some 250 years before Euclid, in the 6th century B.C. The theorem that bears his name is about an equality of non-congruent areas; specifically, the squares that are drawn on each side of a right triangle.

The Pythagorean Theorem

Thus if ABC is a right triangle with the right angle at A, then the square drawn on BC opposite the right angle, is equal to the two squares together drawn on CA, AB.

That is, if it takes one can of paint to paint the square on BC, then it will also take exactly one can to paint the other two squares.

That is how the sides of a right triangle are related — by the squares drawn on them — and we can illustrate it with numbers.

The square drawn on the side opposite the right angle, 25, is equal to the squares on the sides that make the right angle: 9 + 16.  Thus a triangle whose sides are 3-4-5 is right-angled.

That and other facts were known to many cultures long before Pythagoras, but credit has gone to him for being the first to prove the theorem. It is the culmination of Euclid’s first Book.

PROPOSITION 47.  THEOREM


In a right triangle the square drawn on the side opposite the right angle
is equal to the squares drawn on the sides that make the right angle.
 
Let ABC be a right triangle in which CAB is a right angle;
then the square drawn on BC is equal to the two squares on CA, AB.
 
 
On BC draw the square BDEC,
and on BA, AC draw the squares GB, HC;
(I. 46)
 
through A draw AL parallel to BD or CE;
and draw AD and FC.
 
(The proof now shows that the square GB is equal to the rectangle BL, and that the square HC is equal to the remaining rectangle CL.  First, however, the proof demonstrates that triangles FBC, ABD are congruent and therefore equal. Next, the square GB and triangle FBC are on the same base FB and in the same parallels FB, GC; while parallelogram CL and triangle ABD are on the same base BD and in the same parallels BD, AL.)
 
 
Then, since each of the angles BAG, BAC is a right angle, 
GA, CA upon meeting BA make the adjacent angles equal to two right angles;
 
therefore AC is in a straight line with GA. (I. 14)
 
For the same reason, BA is in a straight line with AH.
 
And, since angle DBC is equal to angle FBA, because each is a right angle,
 
to each of them add angle ABC;
 
then the whole angle ABD is equal to the whole angle FBC. (Axiom 2)
 
And since FB is equal to BA, and DB to BC,
 
the two sides FB, BC are equal to the two sides AB, BD respectively;
 
and we proved that angle FBC is equal to angle ABD;
 
therefore triangle FBC is equal to triangle ABD. (S.A.S.)
 
Furthermore, the parallelogram BL is double the triangle ABD, because they are on the same base BD and in the same parallels BD, AL. (I. 41)
 
And the square GB is double the triangle FBC, 
because they are on the same base FB and in the same parallels FB, GC.
But doubles of the equal triangles FBC, ABD are equal to one another.
(I. 37, Problem 2.)
 
Therefore the square GB is equal to the parallelogram BL.
 
In the same way, by drawing AE, BK, we could show that the square HC is equal to the parallelogram CL.
 
Therefore the whole square BDEC is equal to the two squares GB, HC. (Axiom 2)
 
And the square BDEC is drawn on BC, and the squares GB, HC on BA, AC.
 
Therefore the square drawn on BC is equal to the two squares on BA, AC.
 
Therefore, in a right triangle etc.  Q.E.D.

The side opposite the right angle is called the hypotenuse 
(“hy-POT’n-yoos”); which literally means stretching under.

The above proof is Euclid’s, not Pythagoras’s. His proof is believed to have been based on the theory of proportions; Proposition VI. 31.

Now it is also a theorem that if BC is the diameter of a circle and A any point on the circumference, then angle BAC is always a right angle. (Proposition III. 31.) Therefore if we imagine the point A moving along the circumference, then at each point we will have a right angle. The square on BA will grow larger while the square on AC grows smaller, yet they continually adjust themselves so that together they equal the constant square on BC! That equality of areas is what makes Pythagoras’s theorem so remarkable.

Moreover, actually seeing that equality — feeling it, almost — rather than just proving it, is the essence of geometry. For however easy or difficult it might be to prove, actually seeing the fact — that those areas really are equal — remains elusive. Over the centuries therefore there have been many, many proofs of this famous theorem, each one offering a different insight. See Topic 3 of Trigonometry.

The converse

The hypothesis of Proposition 47 is that the triangle is right-angled; hence the converse, which is Proposition 48 and the last theorem of Book I, has for its conclusion that the triangle is right-angled. Here is the enunciation.

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the Half-Life team

Kelly Bailey
Ted Backman
Julie Bazuzi
Yahn Bernier
Ken Birdwell
Steve Bond
Dario Casali
Greg Coomer
Andrew Coward
Wes Cumberland
Ken Eaton
Matt Eslick
John Guthrie
Mona Lisa Guthrie
Mike Harrington
Monica Harrington
Brett Johnson
Erik Johnson
Chuck Jones
Phil Kuhlmey
Marc Laidlaw
Karen Laur
Dave Lee
Miene Lee
Randy Lundeen
Gabe Newell
Yatsze Mark
Lisa Mennet
Cade Myers
Kate Powell
Torsten Reinl
Dave Riller
Jay Stelly
Harry Teasley
Steve Theodore
Bill Van Buren
Douglas R. Wood

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Siris: Login Information

Login Information

Learn ID

Learn ID must be a 8 letters ID used to login to BlackBoard and your Learn email account. To enter your Learn ID, you must first click on the blank textbox so that a blinking cursor will appear. You then enter your 8 letters Learn ID. 

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Your Password is the password used to login to BlackBoard and your Learn email account. To enter your Password, you must first click on the blank textbox so that a blinking cursor will appear. You then enter your Password. In order to keep your Password confidential, each character you enter will display as an asterisk. 

After typing in your Password, click on the Log into SIRIS button. 


Student Advisor Mode

In order to assist students that may be having some difficulties with Siris, the system has a built in Student Advisor Mode feature. This feature will allow authorized employees to log into Siris and view the same information as the student. To do this, you must first get the proper authorization from the Aries Security Administrator in the Registrar’s office. Once proper authorization is obtained and a student comes to you with a Siris problem, you can log into Siris using your e-mail usercode and password (ie. john.smith). You will be greeted with a screen that reads: 

‘You have entered Advisor mode, please enter your College Net password for verification and the number of the student you would like to assist.’ 

Once you have entered Your e-mail password and the student’s ID number, you will be able to view the information as if you were that student. Always remember, when prompted for a password, enter Your e-mail password.

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ENGINEER: HONESTY

HONESTY

Engineers are always honest in matters of technology and human relationships. That’s why it’s a good idea to keep engineers away from customers, romantic interests, and other people who can’t handle the truth.

Engineers sometimes bend the truth to avoid work. They say things that sound like lies but technically are not because nobody could be expected to believe them. The complete list of engineer lies is listed below.

    “I won’t change anything without asking you first.”

“I’ll return your hard-to-find cable tomorrow.”
“I have to have new equipment to do my job.”
“I’m not jealous of your new computer.”

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Q: What kinds of employers are in the service industry?

What kinds of employers are in the service industry? The services-producing sector, which covers a wide range of activities, has been growing rapidly in recent years. The largest industries in this sector are wholesale and retail trade, health care, and social assistance. Some service industries tend to be more regionalized than others. Information about local employers and trends in the local job market can be found at your high school guidance office, campus career centre, youth employment centre, or chamber of commerce. Industry share of services-producing jobs* in Ontario, 2004 (Table content) % by industry Other services (e.g., repair and maintenance, personal, laundry) = 5% Business, building and other support services = 6% Transportation and warehousing = 6% Information, culture and recreation = 7% Public administration = 7% Accommodation and food = 8% Educational services = 8% Finance, insurance, and real estate = 9% Professional, scientific and technical = 9% Health care and social assistance = 14% Wholesale and retail trade = 21% *Jobs are grouped according to the North American Industry Classification System. Note: Percentages are rounded and may not add up to 100. Source: Statistics Canada, Labour Force Survey. (End of table content) What kinds of employers are in the manufacturing industry? Manufacturing industries have the largest share of employment in the goods-producing sector. Transportation equipment (e.g., automotive assembly), chemicals, plastics and rubber products, and food, beverage and tobacco are three of the most important manufacturing industry groupings in Ontario. Different kinds of manufacturing jobs are found in different regions of the province. So before setting a career plan, it's important to know where potential employers are located. Information about employers and trends in the local job market can be found at your high school guidance office, campus career centre, youth employment centre, or local chamber of commerce, or in your local newspaper. Characteristics of the job market Industry share of manufacturing jobs* in Ontario, 2004 (Table content) % by industry Electrical equipment, appliances and components = 3% Textiles, clothing, leather and allied products = 4% Primary metals = 4% Printing and related support activities = 5% Computer and electronic products = 6% Machinery = 7% Wood and paper products = 7% Fabricated metal products = 9% Food, beverage and tobacco products = 10% Chemicals, plastics and rubber products = 12% Other manufacturing = 13% Transportation equipment = 21% *Jobs are grouped according to the North American Industry Classification System. Note: Percentages are rounded and may not add up to 100. Source: Statistics Canada, Labour Force Survey. (End of table content) Matching skills with the needs of employers Some industries hire people with a more specific set of skills than other industries. If you're interested in a particular kind of work, it's important to know which skills employers require for that work. Occupational employment by industry* in Ontario, 2004 Occupational groupings = Management Manufacturing industry = 8% Service industry = 10% Construction industry = 14% Primary industry** 2% Occupational groupings = Business, finance and administration Manufacturing industry = 13% Service industry = 22% Construction industry = 9% Primary industry** = 5% Occupational groupings = Natural and applied sciences Manufacturing industry = 8% Service industry = 7% Construction industry = 2% Primary industry** = 3% Occupational groupings = Health Manufacturing industry –*** Service industry = 7% Construction industry – Primary industry** – Occupational groupings = Social science, education, government service and religion Manufacturing industry – Service industry = 10% Construction industry – Primary industry** – Occupational groupings = Art, culture, recreation and sport Manufacturing industry = 1% Service industry = 4% Construction industry – Primary industry** – Occupational groupings = Sales and service Manufacturing industry = 4% Service industry = 30% Construction industry = 1% Primary industry** – Occupational groupings = Trades, transport and equipment operators Manufacturing industry = 18% Service industry = 9% Construction industry = 73% Primary industry** = 9% Occupational groupings = Occupations unique to primary industry Manufacturing industry – % Service industry 1% Construction industry – Primary industry** 79% Occupational groupings = Processing, manufacturing, and utilities Manufacturing industry = 49%% Service industry = 1% Construction industry – Primary industry** – *Grouped according to North American Industry Classification System. Groupings exclude utilities. **Primary industry includes agriculture, forestry, fishing, mining, and oil and gas. *** “–“ indicates employment of less than 1500. Source: Statistics Canada, Labour Force Survey. (End of table content) Today, many entry-level occupations require a higher level of skills than they did in the past. 

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