## 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

## 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