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Introduction

Discussion : discuss these research topics

Research one of the following geometrical ideas and describe it to your group:

  1. taxicab geometry,
  2. spherical geometry,
  3. fractals,
  4. the Koch snowflake.

Circle geometry

Terminology

The following is a recap of terms that are regularly used when referring to circles.

  • An arc is a part of the circumference of a circle.
  • A chord is defined as a straight line joining the ends of an arc.
  • The radius, r , is the distance from the centre of the circle to any point on the circumference.
  • The diameter is a special chord that passes through the centre of the circle. The diameter is the straight line from a point on the circumference to another point on the circumference, that passes through the centre of the circle.
  • A segment is the part of the circle that is cut off by a chord. A chord divides a circle into two segments.
  • A tangent is a line that makes contact with a circle at one point on the circumference. ( A B is a tangent to the circle at point P ).
Parts of a Circle

Axioms

An axiom is an established or accepted principle. For this section, the following are accepted as axioms.

  1. The Theorem of Pythagoras, which states that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. In A B C , this means that ( A B ) 2 + ( B C ) 2 = ( A C ) 2
    A right-angled triangle
  2. A tangent is perpendicular to the radius, drawn at the point of contact with the circle.

Theorems of the geometry of circles

A theorem is a general proposition that is not self-evident but is proved by reasoning (these proofs need not be learned for examination purposes).

Theorem 1 The line drawn from the centre of a circle, perpendicular to a chord, bisects the chord.

Proof :

Consider a circle, with centre O . Draw a chord A B and draw a perpendicular line from the centre of the circle to intersect the chord at point P . The aim is to prove that A P = B P

  1. O A P and O B P are right-angled triangles.
  2. O A = O B as both of these are radii and O P is common to both triangles.

Apply the Theorem of Pythagoras to each triangle, to get:

O A 2 = O P 2 + A P 2 O B 2 = O P 2 + B P 2

However, O A = O B . So,

O P 2 + A P 2 = O P 2 + B P 2 A P 2 = B P 2 and AP = B P

This means that O P bisects A B .

Theorem 2 The line drawn from the centre of a circle, that bisects a chord, is perpendicular to the chord.

Proof :

Consider a circle, with centre O . Draw a chord A B and draw a line from the centre of the circle to bisect the chord at point P . The aim is to prove that O P A B In O A P and O B P ,

  1. A P = P B (given)
  2. O A = O B (radii)
  3. O P is common to both triangles.

O A P O B P (SSS).

O P A ^ = O P B ^ O P A ^ + O P B ^ = 180 ( APB is a str. line ) O P A ^ = O P B ^ = 90 O P A B

Theorem 3 The perpendicular bisector of a chord passes through the centre of the circle.

Proof :

Consider a circle. Draw a chord A B . Draw a line P Q perpendicular to A B such that P Q bisects A B at point P . Draw lines A Q and B Q . The aim is to prove that Q is the centre of the circle, by showing that A Q = B Q . In O A P and O B P ,

  1. A P = P B (given)
  2. Q P A = Q P B ( Q P A B )
  3. Q P is common to both triangles.

Q A P Q B P (SAS). From this, Q A = Q B . Since the centre of a circle is the only point inside a circle that has points on the circumference at an equal distance from it, Q must be the centre of the circle.

Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
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