Difference between revisions of "1983 AIME Problems/Problem 12"
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== Problem == | == Problem == | ||
The length of diameter <math>AB</math> is a two digit integer. Reversing the digits gives the length of a perpendicular chord <math>CD</math>. The distance from their intersection point <math>H</math> to the center <math>O</math> is a positive rational number. Determine the length of <math>AB</math>. | The length of diameter <math>AB</math> is a two digit integer. Reversing the digits gives the length of a perpendicular chord <math>CD</math>. The distance from their intersection point <math>H</math> to the center <math>O</math> is a positive rational number. Determine the length of <math>AB</math>. | ||
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== Solution == | == Solution == |
Revision as of 02:42, 21 January 2007
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
.