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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

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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 .?
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Stoney Reply
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Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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Damian Reply
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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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Damian Reply
absolutely yes
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
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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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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