document.write( "Question 1100018: The length of the diagonal of a rectangle is shorter than the semi-perimeter by 3/4 length of the shorter side. If the dimensions of all sides are integers, find the minimum length of the longer side. \n" ); document.write( "

Algebra.Com's Answer #714519 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "let x = shorter side
\n" ); document.write( "let y = longer side

\n" ); document.write( "The semiperimeter is then x+y; and then the length of the diagonal is
\n" ); document.write( " $\"%28x%2By%29-%283%2F4%29%28x%29+=+%281%2F4%29x%2B4y\"$

\n" ); document.write( "Then by the Pythagorean Theorem,
\n" ); document.write( " $\"%281%2F4%29x%2By+=+sqrt%28%28x%29%5E2%2B%28y%29%5E2%29\"$
\n" ); document.write( " $\"%281%2F16%29x%5E2%2B%281%2F2%29xy%2By%5E2+=+x%5E2%2By%5E2\"$
\n" ); document.write( " $\"%281%2F2%29xy+=+%2815%2F16%29x%5E2\"$
\n" ); document.write( " $\"8xy+=+15x%5E2\"$
\n" ); document.write( " $\"8y+=+15x\"$ [note dividing by x is okay here, because we know the problem would make no sense if x were equal to 0]

\n" ); document.write( "The smallest solution in integers is y=15, x=8, making the diagonal 17.

\n" ); document.write( "The answer to the problem is the minimum length of the longer side, which is 15. \n" ); document.write( "

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