# Polygons and quadrilaterals  (Page 3/3)

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We use the symbol $|||$ to mean is similar to .

Similar Polygons

Two polygons are similar if:

1. their corresponding angles are equal, and
2. the ratios of corresponding sides are equal.

Show that the following two polygons are similar.

1. We are required to show that the pair of polygons is similar. We can do this by showing that the ratio of corresponding sides is equal and by showing that corresponding angles are equal.

2. We are given the angles. So, we can show that corresponding angles are equal.

3. All angles are given to be 90 ${}^{\circ }$ and

$\begin{array}{ccc}\hfill \stackrel{^}{A}& =& \stackrel{^}{E}\hfill \\ \hfill \stackrel{^}{B}& =& \stackrel{^}{F}\hfill \\ \hfill \stackrel{^}{C}& =& \stackrel{^}{G}\hfill \\ \hfill \stackrel{^}{D}& =& \stackrel{^}{H}\hfill \end{array}$
4. We first need to see which sides correspond. The rectangles have two equal long sides and two equal short sides. We need to compare the ratio of the long side lengths of the two different rectangles as well as the ratio of the short side lenghts.

Long sides, large rectangle values over small rectangle values:

$\begin{array}{ccc}\hfill \mathrm{Ratio}& =& \frac{2L}{L}\hfill \\ & =& 2\hfill \end{array}$

Short sides, large rectangle values over small rectangle values:

$\begin{array}{ccc}\hfill \mathrm{Ratio}& =& \frac{L}{\frac{1}{2}L}\hfill \\ & =& \frac{1}{\frac{1}{2}}\hfill \\ & =& 2\hfill \end{array}$

The ratios of the corresponding sides are equal, 2 in this case.

5. Since corresponding angles are equal and the ratios of the corresponding sides are equal the polygons ABCD and EFGH are similar.

All squares are similar.

If two pentagons ABCDE and GHJKL are similar, determine the lengths of the sides and angles labelled with letters:

1. We are given that ABCDE and GHJKL are similar. This means that:

$\frac{\mathrm{AB}}{\mathrm{GH}}=\frac{\mathrm{BC}}{\mathrm{HJ}}=\frac{\mathrm{CD}}{\mathrm{JK}}=\frac{\mathrm{DE}}{\mathrm{KL}}=\frac{\mathrm{EA}}{\mathrm{LG}}$

and

$\begin{array}{ccc}\hfill \stackrel{^}{A}& =& \stackrel{^}{G}\hfill \\ \hfill \stackrel{^}{B}& =& \stackrel{^}{H}\hfill \\ \hfill \stackrel{^}{C}& =& \stackrel{^}{J}\hfill \\ \hfill \stackrel{^}{D}& =& \stackrel{^}{K}\hfill \\ \hfill \stackrel{^}{E}& =& \stackrel{^}{L}\hfill \end{array}$
2. We are required to determine the

1. $a$ , $b$ , $c$ and $d$ , and
2. $e$ , $f$ and $g$ .
3. The corresponding angles are equal, so no calculation is needed. We are given one pair of sides $DC$ and $KJ$ that correspond. $\frac{DC}{KJ}=\frac{4,5}{3}=1,5$ so we know that all sides of $KJHGL$ are 1,5 times smaller than $ABCDE$ .

4. $\begin{array}{ccc}\hfill \frac{a}{2}=1,5& \therefore & a=2×1,5=3\hfill \\ \hfill \frac{b}{1,5}=1,5& \therefore & b=1,5×1,5=2,25\hfill \\ \hfill \frac{6}{c}=1,5& \therefore & c=6÷1,5=4\hfill \\ \hfill d=\frac{3}{1,5}& \therefore & d=2\hfill \end{array}$
5. $\begin{array}{ccc}\hfill e& =& {92}^{\circ }\left(\mathrm{corresponds to H}\right)\hfill \\ \hfill f& =& {120}^{\circ }\left(\mathrm{corresponds to D}\right)\hfill \\ \hfill g& =& {40}^{\circ }\left(\mathrm{corresponds to E}\right)\hfill \end{array}$
6. $\begin{array}{ccc}\hfill a& =& 3\hfill \\ \hfill b& =& 2,25\hfill \\ \hfill c& =& 4\hfill \\ \hfill d& =& 2\hfill \\ \hfill e& =& {92}^{\circ }\hfill \\ \hfill f& =& {120}^{\circ }\hfill \\ \hfill g& =& {40}^{\circ }\hfill \end{array}$

## Activity: similarity of equilateral triangles

Working in pairs, show that all equilateral triangles are similar.

## Polygons-mixed

1. Find the values of the unknowns in each case. Give reasons.
2. Find the angles and lengths marked with letters in the following figures:

## Investigation: defining polygons

Investigate the different ways of defining polygons. Polygons that you should pay special attention to are:

• Isoceles, equilateral and right-angled triangle
• Kites, parallelograms, rectangles, rhombuses (or 'rhombi'), squares and trapeziums

Things to consider are how these figures have been defined in this book and what alternative definitions exist. For example, a triangle is a three-sided polygon or a figure having three sides and three angles. Triangles can be classified using either their sides or their angles. Could you also classify quadrilaterals in this way? What other names exist for these figures? For example, quadrilaterals can also be called tetragons.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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.
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nanocopper obvius
Alexandre
what is the stm
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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How we are making nano material?
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LITNING
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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what is differents between GO and RGO?
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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
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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
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what is Nano technology ?
write examples of Nano molecule?
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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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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