Thursday, July 24, 2008

Circle Theorems

It is that time of the year again, when I can become very frustrated. Today was my one day off and I had planned to do nothing (which means watch Dr. Who and be a couch potato)! However my Dad happened to ask me to study maths with my ickle (and annoying!) little brother. (I guess the annoying factor runs through the family). Grumbling I muttered that I will do so later, as apart from my dad, no one was too keen on this. Circle theorems was the chosen topic and I had the whole day to do this.

More on what I forgot to do later, but in the book there were three theorems of which I will only mention two.

1) The angle at the centre
The angle subtended at the centre of a circle is double the size of the angle subtended at the edge from the same two points.
So in the above case, a = 2b. You can prove this by simply splitting the above funny shape into two isosceles triangles, and playing around.

2) Angle in a semi-circle
Apologies for not labelling my diagrams, but the angle formed when we join lines from the ends of the diameter to the circumference is 90 degrees.

Sadly in all my childhood the concept of proof was alien-ish and not regarded as important. The challenge used to be in finishing the exercises first. However, today I questioned the second theorem and why we get a right angle. My brother thought I was playing a joke on him, but I was deadly serious when I asked him why the angle in the semi-circle was always 90 degrees.

Having just worked through theorem one, the obvious idea that we came to was that the angle subtended at the centre of the circle happens to be 180 degrees. Hence by theorem one, the angle at the circumference is half of 180, which is 90 degrees. I wonder though, why was this little fact or connection between the theorems not in the (GCSE) book?

You can also prove the second theorem by creating two isosceles triangles again. I may have imagined it, or perhaps I wanted to see my brothers delight at having proved the fact, but there was something or other expressed on his face. I think he tries too hard to not like maths, when he actually likes it. (He's not had the best of teachers).

I was imagining the consequences if he actually went on to study maths, and I decided he is best doing something else. (Namely for my peace of mind!) That being said, he is mathematically more able than I am and I think my dad's idea is a good'un. However he doesn't understand that teaching is not always a liner process, and it is best to first let my brother have the wow-factor for maths. That was my plan when I got him to do the puzzle "Who owns the fish?" which he successfully managed to do. I was delighted at this for I had been unable to solve the puzzle when I was at school. (More reasons for him to study maths I suppose, but you didn't hear anything from me!)

If anyone does know any good site with nice and complex (logic) puzzles, then this bean wants them! He really liked the Fish one and keeps on asking me for more. Well what you rather work on, a logic puzzle or circle theorems?!


Anatoly said...

There are some logic puzzles here:

Beans said...

Thanks Anatoly. Them Tsunami puzzles have been giving me a headache though!

Mission A Star said...

I have done a neat (albeit very familiar) proof of the diameter subtending a right angle in a slightly different style. Check out the Youtube video: