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Mathematics

Grade 9

Numbers

Module 3

Why all this fuss about pythagoras?

INVESTIGATION

1.1 Work in a group but start on your own by drawing three right–angled triangles of different shapes and sizes. Work as accurately as possible. It will be a lot easier if you use squared paper. Now work even more accurately and measure the three sides of each triangle to the nearest millimetre. Complete the first three rows of the table. Now use your calculator to complete the rest of the table.

SYMBOL TRIANGLE A TRIANGLE B TRIANGLE C
Length of the shortest side a .................... .................... ....................
Length of the medium triangle b .................... .................... ....................
Length of the longest side c .................... .................... ....................
Square of the length of the shortest side a 2 .................... .................... ....................
Square of the length of the medium side b 2 .................... .................... ....................
Sum of the two squares above a 2 + b 2 .................... .................... ....................
Square of the length of the longest side c 2 .................... .................... ....................

1.2 There should be something interesting about the shaded cells of the table. In your group write down carefully what you notice and (if you can) why it happens.

2. Take three lines:

  • The three given lines can be used to form a right–angled triangle.

Cutting–out:

2.1 Can the three given squares be used to form a triangle?

3 Write a neat summary of the results of the investigation.

end of INVESTIGATION

The Theorem of Pythagoras goes:

  • In a right–angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.

The importance of the Theorem of Pythagoras is that we use it in two ways: Firstly, if we know that a triangle is right angled, then we can say something very important about its sides. Secondly, if we know that the three sides of a triangle have a certain relationship with each other, then we also know that the triangle must be right–angled.

CLASS WORK

1 We label triangles as follows:

  • Refer to the sketch alongside.
  • The three vertices (corners) get capital letters. (A, B and C).
  • The sides can be named as the two vertices betweenwhich the side lies (AB, BC and AC), or we can uselowercase letters, each corresponding to the vertexopposite the side (a, b and c).
  • At the moment we are dealing with right–angled triangles, but all triangles are labelled the same way.
  • We also use the same letter to refer both to the name of a side and to its length.
  • E.g. PR = 3,5cm or r = 5cm.ΔPRS means: triangle PRS

REMEMBER always to use a ruler for good sketches!

  • In the following exercise the first problem is always an example.

2 Problem : AEH has a right angle at H.AH = 6cm and EH = 8cm.Draw a sketch (not accurate) of the triangle and use the Theorem of Pythagoras to calculate the length of side AE.

Solution : Because we know that the triangle has a rightangle, we are allowed to say that AE 2 = AH 2 + EH 2 (or: h 2 = e 2 + a 2 )

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
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Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
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Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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Cied
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abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
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Source:  OpenStax, Mathematics grade 9. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col11056/1.1
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