PMC 15th Presentation

By

Sadiq Merchant

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Pythagorean Theorem and Life – An Online Research

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Three main objectives of this online research are: Trace the history of Pythagorean TheoremDiscuss few related teaching ideasShare certain real life applications of Pythagorean Theorem

Objectives

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The Pythagorean Theorem is a special rule in mathematics and its name was probably kept after the ancient Greek mathematician Pythagoras (569-500 B.C.), who was perhaps the first to offer a proof of the theorem.

But the fact is people had noticed the special relationship between the sides of a right triangle long before that.

History

The Egyptians knew about the right angled triangle.

They were able to use this knowledge in the construction of pyramids, temples and other buildings.

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Egyptians: Pyramids

http://www.blog.dolorescannon.com/wp-content/uploads/2013/04/Egyptian-pyramid-and-stars.jpg

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Ancient clay tablets from Babylonia indicate that they had rules for generating ‘Pythagorean triples’ about 1000 years before Pythagoras

This fact indicates that they understood the relationship between the sides of a right triangle.

(Illustration source: http://www.swan.ac.uk/compsci/ResearchGroups/TheoryGroups/AlgMethFolder/DSTFolder/HistoryOfTables/Plimpton/Plimpton1TN.GIF)

Babylonian Clay Tablets

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An ancient Chinese treatise called Chou Pei Suan Ching; predating Pythagoras, gives a statement and a demonstration of the Theorem.

http://saxakali.com/COLOR_ASP/developcm3.htm

Chou Pei Suan Ching

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Ancient mathematicians from the Indian sub-continent also knew the Pythagorean theorem, in the form of Sulba-sutras (800-600 B.C.) These Sutras discuss the strict requirements for the orientation, shape, and area of altars for religious purposes. http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_sulbasutras.html

Ancient Indian Mathematicians

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It has also been suggested that the ancient Mayans also used variations of Pythagorean triples in their Long Count Calendar.

Mayan’s Long Count calendar

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For a right angled triangle with sides a, b, c the Pythagorean Theorem states that the sum of the squares of the lengths of the two shorter sides of any right angled triangle will always be equal to the square of the length of the longest side. (Illustration source: https://support.groundspeak.com/index.php?

pg=file&from=2&id=95)

The Pythagoras Theorem

For Whole No. values One of the legs of the

right triangle is a multiple of 3.

While the same leg or the other one is a multiple of 4.

One of the three sides is a multiple of 5.

(3,4,5)(5,12,13)(7,24,25)(8,15,17)(9,40,41)(11,60,61)(12,35,37)(13,84,85)(16,63,65)

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Some Interesting Observation

Visual Proof

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Pythagoras Theorem

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Proof: Pythagorean Theorem

Step 1: Cut four identical right triangles from a piece of paper.

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A simple hands on proof

a

b

c

A simple hands on proof

Step 2: Arrange the triangles with the hypotenuse of each forming a square.

a

bc

b

b

a

a

b

c

c

ca

Area of large square = (a + b)2

Area of each part4 Triangles = 4x½ab = 2abRed Square = c2

Total Area of 4 Triangles and Red Square = 2ab +c2

Thus; (a + b)2 = 2ab +c2

a2 + 2ab + b2 = 2ab +c2

a2 + b2 = c2

100 plus ways to prove Pythagoras Theorem

http://www.cut-the-knot.org/pythagoras/

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Some More Proofs

Draw a square on each side.

Another Hands’ on Activity

A Pythagorean Puzzle

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1

2


1

2

3


1

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4 A Pythagorean Puzzle

1

23

4


1

23

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1

23

4

5


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The Pythagorean theorem has far-reaching implications in many fields:

Computing distances between two points, such as in navigation and land surveying.

Finding area of triangular surfaces.Designing ramps for handicap persons,

skateboard parks.Evaluating rational numbers.

Applications

Suppose you drive directly west for 48 miles,

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“Distance Problem”

48

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Then turn south and drive for 36 miles.


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36?

How far are you from where you started?


c

On Applying Pythagorean Theorem

482 362+ = c2

12962304 c2

3600 2c

60 = cSo you are 60 miles away from where you started

482 362+ = c2

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(x1,y1)

(x2,y2)a = x2-x1

b = y2-y1c = distance

The Pythagorean Theorem is often easier for students to learn than the distance formula.

The Distance Formula

Find the distance between two points, a and b with the given co-ordinates a(3, 4) & b(-4, 1)

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a

b3

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The Distance Formula

2 2 23 7ab 2 23 7ab

7.6 (1 )ab dp approx.

Given that the base of the triangle is 10 meters, and other two sides are of 7 meters.

The Polygon is divided into two triangles each having height “h”

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Area of an Isosceles Triangle

Using the Pythagorean Theorem the value of h is found.

h

10 m

7 m7 m

72 = 52 + h2

49 = 25 + h2

24 = h2

h = √24 Δ = ½ bh

= ½ (10)(√24)

≈ 24.5 m2

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Area of an Isosceles Triangle

5 m 5 m

h

10 m

7 m7 m

h

10 m

7 m7 m

1m

8m

Length of the Ramp

What is the length of the slope?

1m

8m

c

b=

a=

c²=a²+ b²

c²=1²+ 8²

c²=1 + 64

c²=65

On Applying Pythagoras’ Theorem

So c = √65 = 8.1 m approx.

Value of Irrational Numbers

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1

2

Unit Square

Using a unit square, the approximate value of the irrational number 2 was evaluated.

1

1

2

1On continuing the diagram value of the irrational number 3 was evaluated.Continuing the diagram 3, 5, 6, 7, etc. were also evaluated

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Value of Irrational Numbers

1

1

2

1

3

1

41

51

6

17

1

8

1

9

1

10

1

11

1

12

1

13

1

14

1

15 1

16

117

1

18

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Some Interesting Application

The skyscrapers shown above are connected by a skywalk with support beams. Here Pythagorean Theorem can be applied to find the length of each support beam.

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Some Interesting Application

x2 = (23.26)2 + (47.57)2

x2 = 2803.9325x ≈ 52.95

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3D Problems: Pythagorean Theorem

A B

E FC D

G H

3 cm

5 cm

12 cm

The diagram shows a rectangular box with top ABCD & base EFGH. Find the distance BG

Finding FG, ΔGHF FG2 = 52 + 122

FG = (52 + 122)FG = 13 cm

13 cm

Finding BG. ΔGFB BG2 = 32 + 132

FG = (32 + 132)FG = 13.3 cm

13.3 cm

A person wanted to buy a 21 inch T.V. He found a TV that was 16 inch high and 14 inch wide. Has he found the TV which he was looking for?

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What Size TV Should You Buy?

W2 = (16)2 + (14)2

x2 = 256 + 196x ≈ 21.26 approx.

http://www.contracosta.edu/math/pythagoras.htm http://www.cut-the-knot.org/ http://www.contracosta.edu/math/pythagoras.htm http://www.ies.co.jp/math/java/geo/pythafv/yhafv.html http://jwilson.coe.uga.edu/emt669/Student.Folders/Huberty.Greg/Pythagorean.html http://www.cut-the-knot.org/pythagoras/ http://www.contracosta.edu/math/pythagoras.htm http://www.cut-the-knot.org/ http://www.contracosta.edu/math/pythagoras.htm

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References

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Sadiq Merchant

[email protected]

Thanks for listening patiently.

PMC 15th Presentation

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