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Circles iv

  1. Find the values of the unknown letters.

Theorem 9 Two tangents drawn to a circle from the same point outside the circle are equal in length.

Proof :

Consider a circle, with centre O . Choose a point P outside the circle. Draw two tangents to the circle from point P , that meet the circle at A and B . Draw lines O A , O B and O P . The aim is to prove that A P = B P . In O A P and O B P ,

  1. O A = O B (radii)
  2. O A P = O P B = 90 ( O A A P and O B B P )
  3. O P is common to both triangles.

O A P O B P (right angle, hypotenuse, side) A P = B P

Circles v

  1. Find the value of the unknown lengths.

Theorem 10 The angle between a tangent and a chord, drawn at the point of contact of the chord, is equal to the angle which the chord subtends in the alternate segment.

Proof :

Consider a circle, with centre O . Draw a chord A B and a tangent S R to the circle at point B . Chord A B subtends angles at points P and Q on the minor and major arcs, respectively. Draw a diameter B T and join A to T . The aim is to prove that A P B ^ = A B R ^ and A Q B ^ = A B S ^ . First prove that A Q B ^ = A B S ^ as this result is needed to prove that A P B ^ = A B R ^ .

A B S ^ + A B T ^ = 90 ( TB SR ) B A T ^ = 90 ( 's at centre ) A B T ^ + A T B ^ = 90 ( sum of angles in BAT ) A B S ^ = A B T ^ However, AQB ^ = A T B ^ ( angles subtended by same chord AB ) A Q B ^ = A B S ^ S B Q ^ + Q B R ^ = 180 ( SBT is a str. line ) A P B ^ + A Q B ^ = 180 ( ABPQ is a cyclic quad ) S B Q ^ + Q B R ^ = A P B ^ + A Q B ^ AQB ^ = A B S ^ A P B ^ = A B R ^

Circles vi

  1. Find the values of the unknown letters.

Theorem 11 (Converse of [link] ) If the angle formed between a line, that is drawn through the end point of a chord, and the chord, is equal to the angle subtended by the chord in the alternate segment, then the line is a tangent to the circle.

Proof :

Consider a circle, with centre O and chord A B . Let line S R pass through point B . Chord A B subtends an angle at point Q such that A B S ^ = A Q B ^ . The aim is to prove that S B R is a tangent to the circle. By contradiction. Assume that S B R is not a tangent to the circle and draw X B Y such that X B Y is a tangent to the circle.

A B X ^ = A Q B ^ ( tan - chord theorem ) However , ABS ^ = A Q B ^ ( given ) A B X ^ = A B S ^ But , ABX ^ = A B S ^ + X B S ^ can only be true if , XBS ^ = 0

If X B S ^ is zero, then both X B Y and S B R coincide and S B R is a tangent to the circle.

Applying theorem [link]

  1. Show that Theorem [link] also applies to the following two cases:

B D is a tangent to the circle with centre O . B O A D . Prove that:
  1. C F O E is a cyclic quadrilateral
  2. F B = B C
  3. C O E / / / C B F
  4. C D 2 = E D . A D
  5. O E B C = C D C O

  1. F O E ^ = 90 ( BO OD ) F C E ^ = 90 ( subtended by diameter AE ) C F O E is a cyclic quad ( opposite 's supplementary )
  2. Let O E C ^ = x .

    F C B ^ = x ( between tangent BD and chord CE ) B F C ^ = x ( exterior to cyclic quad CFOE ) B F = B C ( sides opposite equal 's in isosceles BFC )
  3. C B F ^ = 180 - 2 x ( sum of 's in BFC ) O C = O E ( radii of circle O ) E C O ^ = x ( isosceles COE ) C O E ^ = 180 - 2 x ( sum of 's in COE )
    • C O E ^ = C B F ^
    • E C O ^ = F C B ^
    • O E C ^ = C F B ^
    C O E ||| C B F ( 3 's equal )
    1. In E D C

      C E D ^ = 180 - x ( 's on a str. line AD ) E C D ^ = 90 - x ( complementary 's )
    2. In A D C

      A C E ^ = 180 - x ( sum of 's ACE ^ and ECO ^ ) C A D ^ = 90 - x ( sum of 's in CAE )
    3. Lastly, A D C ^ = E D C ^ since they are the same .

    4. A D C ||| C D E ( 3 's equal ) E D C D = C D A D C D 2 = E D . A D
    1. O E = C D ( OEC is isosceles )
    2. In B C O

      O C B ^ = 90 ( radius OC on tangent BD ) C B O ^ = 180 - 2 x ( sum of 's in BFC )
    3. In O C D

      O C D ^ = 90 ( radius OC on tangent BD ) C O D ^ = 180 - 2 x ( sum of 's in OCE )
    4. Lastly, O C is a common side to both 's.

    5. B O C ||| O D C ( common side and 2 equal angles ) C O B C = C D C O O E B C = C D C O ( OE = CD isosceles OEC )

F D is drawn parallel to the tangent C B Prove that:
  1. F A D E is cyclic
  2. A F E ||| C B D
  3. F C . A G G H = D C . F E B D

  1. Let B C D = x

    C A H = x ( between tangent BC and chord CE ) F D C = x ( alternate , FD CB ) FADE is a cyclic quad ( chord FE subtends equal 's )
    1. Let F E A = y

      F D A = y ( 's subtended by same chord AF in cyclic quad FADE ) C B D = y ( corresponding 's, FD CB ) F E A = C B D
    2. B C D = F A E ( above )
    3. A F E = 180 - x - y ( 's in AFE ) C B D = 180 - x - y ( 's in CBD ) A F E ||| C B D ( 3 's equal )
    1. D C B D = F A F E D C . F E B D = F A
    2. A G G H = F A F C ( FG CH splits up lines AH and AC proportionally ) F A = F C . A G G H
    3. F C . A G G H = D C . F E B D

Questions & Answers

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
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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.
Damian
yes that's correct
Professor
I think
Professor
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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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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