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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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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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