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Mathematics

Grade 9

Algebra and geometry

Module 10

Similarity

ACTIVITY 1:

To practically investigate the conditions for similarity

1. The pentagons ABDEF and LCMRK are given (A-6). LCMRK is an enlargement of ABDEF. What is the scale factor by which ABDEF were enlarged to give LCMRK?

2. Write down the ratios between the corresponding pairs of sides of ABDEF and LCMRK.

3. Write down the relationship between the corresponding pairs of angles of the two figures.

4. These two figures are not congruent. What do we call them?

5. Name as many as possible e x amples of this phenomenon in real life.

Similar figures:

  • The pentagons in the activity above are similar. They have the same form, but do not have the same size.
  • Their corresponding angles have the same magnitudes.
  • Their corresponding sides are in the same ratio.

Therefore LK AF = KR FE = MR DE = CM BD = CL BA = 3 1 size 12{ { { ital "LK"} over { ital "AF"} } = { { ital "KR"} over { ital "FE"} } = { { ital "MR"} over { ital "DE"} } = { { ital "CM"} over { ital "BD"} } = { { ital "CL"} over { ital "BA"} } = { {3} over {1} } } {} This constant ratio also is the scale factor of the enlargement.

  • We say that ABDEF  LCMRK. Note that the order of the letters is in the same order of the angles which are equal and the sides which are in proportion. (The symbol for similarity is )

Homework assignment

1. Measure the lengths of the sides and the magnitudes of the angles in the following figures (A-7) and decide whether they are similar or not. If the two figures are not similar, give a reason why they are not similar.

2. If the corresponding a ngles of two quadrilaterals are equ a l , are they necess a rily also simil a r ?

3. If corresponding sides of two quadrilaterals are proportion a l , are they necess a rily also simil a r ?

  • In the homework assignment above you saw that, for quadrilaterals to be similar, both conditions of similarity must be satisfied. In other words, the corresponding angles must be equal and the corresponding sides must be proportional. Do the same conditions also apply to triangles?

ACTIVITY 2:

To practically investigate the conditions for similarity in triangles

[LO 3.5]

Construct ΔABC and ΔDEF. Calculate the magnitudes A and E.

1.2 Are the corresponding angles of the two triangles equal?

1.3 Complete the following:

AB ED = size 12{ { { ital "AB"} over { ital "ED"} } ={}} {} ....................

BC DF = size 12{ { { ital "BC"} over { ital "DF"} } ={}} {} ....................

AC EF = size 12{ { { ital "AC"} over { ital "EF"} } ={}} {} ....................

1.4 Are the corresponding sides of the two triangles proportional?

1.5 Are the two triangles similar?

1.6 Complete the following: If the corresponding angles of two triangles are equal, their corresponding sides are necessarily also always ......................... This means that, if the corresponding angles of triangles are equal the triangles are .........................

2.1 Construct the following two triangles:

2.2 Are the sides of the two triangles proportional?

2.3 Measure all the angles of ΔABC and ΔMOR. What do you find?

2.4 Is ΔABC  ΔMOR?

2.5 Complete the following: If the corresponding sides of two triangles are proportional then their corresponding ..................................... are equal. That therefore means that, if the corresponding sides of two triangles are proportional, the triangles are.....................................

  • We therefore see that with triangles only one of the conditions of similarity have to be present for triangles to be similar.
  • That means that, if the three a ngles of one tri a ngle are equal to the three a ngles of the other tri a ngle , then the corresponding sides of the two triangles are proportional and the triangles are therefore also similar.
  • It also means that, if the corresponding sides of the triangles are proportion a l , then the corresponding angles of the two triangles are equal and the triangles are therefore also similar.

Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
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
characteristics of micro business
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
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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