1 CT PPT How Zhao Yin Ho Qi Yan Ong Ru Yun Pythagoras and the Pythagorean Theorem 2014

NAME : HOW ZHAO YIN (201420026)

HO QI YAN (201420009)

ONG RU YUN (201420030)

CLASS : FOUNDATION IN LIBERAL ARTS

MODULE : INTRODUCTION TO CRITICAL THINKING

LECTURER : MS. DOT MACKENZIE

TERM : MAY 2014

DATE : 14 JULY 2014

TOPIC : PYTHAGORAS AND THE PYTHAGOREAN THEOREM

PYTHAGORAS AND THE PYTHAGOREAN THEOREM

Illustration source: http://www.edb.utexas.edu/visionawards/petrosino/Media/Members/zhfbdzci/pythagoras1.gif

According to the UALR (2001), “The Pythagorean theorem takes its name from the ancient Greek mathematician Pythagoras (569 B.C.-500 B.C.), who was perhaps the first to offer a proof of the theorem. But people had noticed the special relationship between the sides of a right triangle long before Pythagoras.”

WHERE WAS PYTHAGORAS

BORN?

Samos

Illustration source: http://intmstat.com/blog/2008/03/samos.jpg

WHAT IS PYTHAGOREAN THEOREM?

According to the UALR (2001),

“The Pythagorean theorem states that the sum of the squares of the lengths of the two other sides of any right triangle will equal the square of the length of the hypoteneuse, or, in mathematical terms, for the triangle shown at right, 

a2 + b2 = c2.

Integers that satisfy the conditions

a2 + b2 = c2 

are called "Pythagorean triples." ”

RIGHT-ANGLED TRIANGLEIllustration source: http://www.mathopenref.com/images/triangle/hypotenuse.gif

Illustration source: http://www.mathsaccelerator.com/measurement/images/triangle-answer.gif

Right-Angled Triangle

HOW TO PROVE THE EQUATION OF PYTHAGOREAN

THEOREM?

cb

a

•There are four similar triangle with the rotation of different angle which are 90°, 180°, and 270°.•Area of triangle can be calculated by using this formulae:

½ x a x b

•The four triangles combined together to form a square shape with a square hole.•The length of side of square inside is a-b.•The area of square inside is (a-b)² or 2ab.•The area of four triangles is 4(½ x a x b).In the last, we get this formulae

c²= (a - b)² + 2ab  = a² - 2ab + b² + 2ab  = a² + b²

Illustration source: http://cdn.instructables.com/FN4/7VG4/GVZPOZOZ/FN47VG4GVZPOZOZ.MEDIUM.gif

Video source: http://www.youtube.com/watch?v=hTxqdyGjtsA&feature=related

EXERCISE 1:

Prove triangle X is a right-angled triangle.

http://fc05.deviantart.net/fs70/f/2013/297/b/1/simple_background_by_biebersays-d6rnj7n.jpg

SOLUTIONS: c2= b2+a2

Let AC2=AB2+BC2

AB2+BC2=82+152

AC2=64+225 =289 √AC2=√289 AC=17 cm

http://fc05.deviantart.net/fs70/f/2013/297/b/1/simple_background_by_biebersays-d6rnj7n.jpg

EXERCISE 2:

Assuming that triangle Q is a right-angled triangle, find the length of side

YZ.

http://hqwide.com/minimalistic-multicolor-gaussian-blur-simple-background-white-wallpaper-5602/

SOLUTIONS: c2=b2+a2

Let ZY2=ZX2+XY2

ZX2+XY2=122+52

ZY2=144+25

=169

√ZY2=√169

ZY=13 cm

http://hqwide.com/minimalistic-multicolor-gaussian-blur-simple-background-white-wallpaper-5602/

EXERCISE 3:State whether the given triangle is a Pythagorean Triple. Give a reason for

your answer.

(12, 35, 37)

http://hqwide.com/gaussian-blur-gradient-simple-background-blurred-colors-wallpaper-62699/

122+352=144+1225

=1369

372=1369

122+352=372, therefore it has been proved that 12, 35, 37 are the sides of a Pythagorean

Triangle.

SOLUTIONS:

http://hqwide.com/gaussian-blur-gradient-simple-background-blurred-colors-wallpaper-62699/

EXERCISE 4:

The legs of a right triangle

are consecutive positive

integers. The hypotenuse has

length 5 cm. What are the

lengths of the legs?

http://www.wallsave.com/wallpaper/1920x1080/simple-light-gradient-211999.html

c2=b2+a2

52=b2+a2

Let a=b-1, b=a+1

52=b2+(b-1)2

52=b2+(b-1)(b-1)

52=b2+b2-b-b+1

25=2b2-2b+1

÷2 0=2b2-2b-24

0=b2-b-12

0=(b+3)(b-4)

Hypotenuse b+3=0 b=-3b should be positive, therefore b=-3 is not acceptable.

b-4=0 b=4 cma+1=ba+1=4 a=3 cm

Therefore, lengths of legs=3 cm, 4 cm

http://www.wallsave.com/wallpaper/1920x1080/simple-light-gradient-211999.html

SOLUTIONS:

REFERENCESBogomolny, A. (2012). Pythagorean Theorem. Retrieved July 9, 2014,

from Cut The Knot: http://www.cut-the-knot.org/pythagoras/.

Section 9.6 The Pythagorean Theorem. (2007). Retrieved July 10,

2014, from Msenux Redwoods:

http://msenux.redwoods.edu/IntAlgText/chapter9/section6solutions.pdf.

Smoller, L. (2001, May). The History of Pythagorean Theorem.

Retrieved July 10, 2014, from UALR College of Information Science and

Systems Engineering: http://ualr.edu/lasmoller/pythag.html.

ThanksFor

Listening

http://www.ucsa.nl/wp-content/uploads/2012/10/Questions-and-Answers.jpeg

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