document.write( "Question 262758: a rectangle has length a, width b and diagonal c. All three are integers less than 100, c is 40 units less than the sum of a and b, and they have no common factors. find a, b, c \n" ); document.write( "

Algebra.Com's Answer #193535 by drk(1908) $\"\"$ $\"About$

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FRom the given, we know that
\n" ); document.write( " $\"a%5E2+%2Bb%5E2+=+c%5E2\"$
\n" ); document.write( "we also now that c= a+b - 40
\n" ); document.write( "By substitution, we get
\n" ); document.write( " $\"a%5E2+%2B+b%5E2+=+%28a%2Bb-40%29%5E2\"$
\n" ); document.write( "expanding the right side, we get
\n" ); document.write( " $\"a%5E2+%2B+b%5E2+=+a%5E2+%2B+b%5E2+%2B+1600+%2B2ab+-80a+-+80b\"$
\n" ); document.write( "subtracting a^2 and b^2 from both sides, we get
\n" ); document.write( " $\"0+=+1600+%2B+2ab+-+80a+-+80b\"$
\n" ); document.write( "or
\n" ); document.write( " $\"80a+%2B+80b+-+2ab+=+1600\"$
\n" ); document.write( "we can solve for a in terms of b. factor out 2a to get
\n" ); document.write( " $\"2a%2840+-b%29+%2B+80b+=+1600\"$
\n" ); document.write( "subtract 80b to get
\n" ); document.write( " $\"%282a%29%2840-b%29+=+1600+-+80b\"$
\n" ); document.write( "divide by 2(40-b) to get
\n" ); document.write( " $\"a+=+%2880%2820-b%29%29%2F%282%2840-b%29%29\"$
\n" ); document.write( "which simplifies to
\n" ); document.write( " $\"a+=+%2840%2A%2820-b%29%29+%2F+%2840-b%29\"$
\n" ); document.write( "At this point we know that b <= 20 OR b > 40.
\n" ); document.write( "Since we want integers for a and b, here are a few options in (a,b) form:
\n" ); document.write( "(8,15) or (15,8)
\n" ); document.write( "If we use
\n" ); document.write( "(8,15), then c = 17 and all 3 share no common factors. But this doesn't satisfy our restrictions.
\n" ); document.write( "Our only new options for (a,b,c) are:
\n" ); document.write( "(67, 72, 97) \n" ); document.write( "

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