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°.



example of reflex angle at centre O, what is the value of $ \angle $ at Circumference point A?

7. 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