1983 AIME Problems/Problem 12
Problem
The length of diameter is a two digit integer. Reversing the digits gives the length of a perpendicular chord
. The distance from their intersection point
to the center
is a positive rational number. Determine the length of
.
Solution
Let and
. It follows that
and
. Applying the Pythagorean Theorem on
and
,
.
Because is a positive rational number, the quantity
cannot contain any square roots. Therefore,
must equal eleven and
must be a perfect square (since
). The only pair
that satisfies this condition is
, so our answer is
.