Brain Teasers
Circle Segments
In the diagram at left, three different line segments each divide a quarter-circle
into two regions of equal area. Rank those three segments from shortest to
longest.
This brainteaser was
written by Derrick Niederman.
Solution:
middle, shortest, longest.
The length of the segment on the left is simply the radius
of the circle, and the segment in the second circle is clearly shorter than the
radius.
So the question is, is the segment in the third circle
longer or shorter than the other two? If you think of the segment in the third
circle as the diagonal of a square, that square must extend beyond the circle
for the segment to bisect the area. Consequently, the diagonal must be greater
than the radius of the circle, so the third segment is the longest.
But perhaps that picture doesn’t convince you. A more
rigorous proof is to express the length of the segment in terms of the radius.
Let x represent the lengths of the
legs of the triangle, and h is the
length of the segment in question.
The area of the triangle is 
. But
its area is also 
, because the triangle is
one of two equal halves in the quarter‑circle. This leads to an equation that
can be solved for x in terms of r.
The hypotenuse of the triangle, h, is the segment about which we are concerned. Its length can now
be found with the Pythagorean theorem.
Because h > r, the third segment is greater than the
radius, so it is the longest.
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