The Pythagorean Theorem
reproduction of Volume 2,
Three Skills for Algebra, Chapter 17
The Pythagorean theorem is one of the oldest statements in mathematics.
This theorem is used to recognize right triangles in the plane and also
to compute the distance between points in the plane. There are hundreds
of proofs of this theorem. One that seems easiest to follow is the
so-called Chinese square (dissection) proof.
Theorem 17.1 [Pythagorean Theorem] If a right triangle has a
hypotenuse of length c and other two sides of lengths a and
b,
then
The proof below employs the identity (a+b)2 = a2+
2ab + b2
|
Geometric Demonstration of (a+b)2 =
a2+ 2ab + b2
for the case where a and b are positive.
First, observe the area of the large square is
(a+b)2
Second, observe the area is of square is the sum of the areas
of four small rectangles
: a2+ 2ab + b2
Third, two difference ways to calculate the areas of the
square must give the same result. Thus the two formulas must
give the same value:
(a+b)2 = a2+ 2ab + b2
|
The Chinese square proof of the Pythagorean theorem is given next. The
proof is based on the construction and then dissection or division of a
large square into a smaller square plus four corner triangles. This
division gives a second way of computing the area of the large square.
The first way is to compute the square L2 of the length
L of a side. The proof of the Pythagorean theorem follows by
equating the formulas for the two different ways and then simplifying the
resulting equation. Details follow.
Chinese Square Proof: To see why
a2+b2 = c2, first
draw a square with four sides of length b+a as follows:
The area of this square is
Second, put four copies of the original right triangle into the
square in the following manner:
Next
Similarly the interior angles of the region, a quadrilateral,
bounded by the four hypotenuses are all equal to g = 90°. The
hypotenuses of the four copies of the original right triangle in the corner
of the square of sides a+b also have a common length
c. Thus they form sides of a square. Finally observe, the area
A of the larger square of sides a+b is also given by
the area of the interior square plus the areas of the four corner copies.
Now from the previous steps, we have A =
c2+4·[(ab)/2] = c2+2ab.
From this a2+2ab+b2 = A
= c2+2ab. Cancellation now implies the
Pythagorean equality a2+b2 =
c2 holds.
Question: Suppose the three sides of a triangle have lengths
a, b and c satisfying
a2+b2 = c2. Is
the triangle a right triangle? (The answer is yes. One way to see why
requires the cosine law in trigonometry - a subject for further inquiry
perhaps.)
|
|
Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
|
|