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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


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


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

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.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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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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