Depending on individuals’ experiences, the following might not (and hope not) happen to you. However, it did happen to some (including me).
When I first studied the topic on “central angle + arc length (also the area of a sector but skipped)”, my teacher did not give a rigorous proof on it (probably because of the question stated below). He just pointed out that “you can see for yourself that the wider is the central angle, the longer is the corresponding arc”. We were made to believe that “the length of the arc is proportional to the central angle subtended”.
The first question I want to ask is:- what method can we use to prove the above, using high-school level language?
Now, for those who accept such finding by observation, how about if I say “see for yourself that the wider is the central angle, the longer is the corresponding chord”. This further implies “the length of the chord is proportional to the central angle subtended”.
The second question is:- If the last remark, based on the second observation, is not true (and in fact it is NOT), why should we believe that from the first observation is true?
Best Answer
You know that the perimeter of the circle is $2\pi R$, with $R$ being the radius.
This is the lenght of the arc corresponding to angle $2\pi$.
The length of the arc corresponding to an angle of $\theta$, with $0<\theta<2\pi$ is:
$L=\theta R$
EDIT: of course this is a formula. However you can relatively easily see that half the angle leads to half the length, and also a quarter, for simple basic symmetry. Thus the proportionnality.