Friday, December 5, 2014

GEOGEBRA angle at centre equal twice angle at circumference

GEOGEBRA angle at center equal twice angle at circumference
customized from https://www.geogebratube.org/material/show/id/37863 by damienchew

http://tube.geogebra.org/student/m359109

students must be able to understand why $ \angle $ at Centre = 2 times $ \angle $  at Circumference.

Steps:


  1. Compare angles at the centre of a circle with angle touching the circumference.
  2. vary the $ \angle $ at Centre O for which it is acute less than 90 °
    example of acute angle at centre O, what is the value of $ \angle $ at Circumference point A?
  3. write down the value of  $ \angle $ at Centre O and $ \angle $ at Circumference point A
  4. vary the $ \angle $ at Centre O for which it is obtuse more than 90° and less than 180°.
    example of obtuse angle at centre O, what is the value of $ \angle $ at Circumference point A?
  5. do step 3
  6. vary the $ \angle $ at Centre O for which it is reflex more than 180°.
  7. example of reflex angle at centre O, what is the value of $ \angle $ at Circumference point A?
  8. do step 3

Thinking:

looking at the evidence of the table of recorded values, suggest a relationship between 
$ \angle $ at Centre O and $ \angle $ at Circumference point A.


Conclusion:

$ \angle $ at Centre = 2 times $ \angle $ at Circumference.


Proof:


Let $ \angle $AOC = 2a

Let $ \angle $BOC = 2b

Then $ \angle $AOB = 360° - 2a – 2b

$ \angle $ OCA = 90° – a (isosceles triangle)

$ \angle $BCO = 90° – b (isosceles triangle)

Therefore, $ \angle $ACB = (90° – a) + (90° – b) =  180° – a – b

Hence, $ \angle $AOB = 2$ \angle $ACB ($ \angle $ at Centre = 2 times $ \angle $ at Circumference) Proven