SOLUTION: 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 t

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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.
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
The area of the triangle is A=%281%2F2%29a%2Ab=%281%2F2%29a%2A6+=+3a.
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.
By the Pythagorean theorem,
c%5E2=a%5E2%2Bb%5E2,
%282a-6%29%5E2+=+a%5E2+%2B+6%5E2,
4a%5E2+-+24a+%2B36+=+a%5E2+%2B+36,
3a%5E2-24a+=+0,
a%5E2+-+8a+=+0, or
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.