Expert: Ahmed Salami - 5/31/2004

Question
Prove that a straight line drawn from the centre of a circle perpendicular to a chord will, if extended, bisect the arc cut off by this chord.


Answer
Hi Matt,
A straight line drawn from the centre of a circle would only be perpendicular to a chord if it bisects the chord.
This line also bisects the angle formed by the lines drawn from the centre of the circle to the ends of the chord. If the line is extended up to the circumference of the circle, then it would be equal in length to the the other two lines which are radii. There are therefore two identical sectors on either side of this line. These sectors are equal in every respect including their arc length.
You can also imagine the segment formed by the chord to be cut off and it would be possible to fold it into two halves along the traces of this line. This also means equal arc lengths.