1983 AIME Problems/Problem 12
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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
.
See also
1983 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |