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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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
is it a question of log
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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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