document.write( "Question 355054: this is the problem. A right triangle has two legs with lengths a and b and a hypotenuse with length c. in this triangle, the area and perimeter are the same nonzero number. find the length of a if b=6. \n" ); document.write( "

Algebra.Com's Answer #253599 by robertb(5830) $\"\"$ $\"About$

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The area of the triangle is $\"A=%281%2F2%29a%2Ab=%281%2F2%29a%2A6+=+3a\"$ .
\n" ); document.write( "The perimeter, on the other hand, is $\"P=+a%2Bb%2Bc+=+a%2B6%2Bc\"$ . Since area = perimeter, then $\"3a+=+a%2B6%2Bc\"$ , or $\"2a+=+6%2Bc\"$ , or $\"c+=+2a+-+6\"$ .
\n" ); document.write( "By the Pythagorean theorem,
\n" ); document.write( " $\"c%5E2=a%5E2%2Bb%5E2\"$ ,
\n" ); document.write( " $\"%282a-6%29%5E2+=+a%5E2+%2B+6%5E2\"$ ,
\n" ); document.write( " $\"4a%5E2+-+24a+%2B36+=+a%5E2+%2B+36\"$ ,
\n" ); document.write( " $\"3a%5E2-24a+=+0\"$ ,
\n" ); document.write( " $\"a%5E2+-+8a+=+0\"$ , or
\n" ); document.write( " $\"a%2A%28a-8%29+=+0\"$ . Therefore $\"a+=+0\"$ or $\"a+=+8\"$ . Discard the first value, so $\"a+=+8\"$ . The hypotenuse is then $\"c+=+2%2A8-6+=+10\"$ . \n" ); document.write( "

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