<< Chapter < Page Chapter >> Page >

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

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
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
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how to synthesize TiO2 nanoparticles by chemical methods
Zubear
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Siyavula textbooks: grade 12 maths' conversation and receive update notifications?

Ask