Congruent TrianglesDate: 6/26/96 at 8:21:52 From: Anonymous Subject: Congruent Triangles Dr. Math, If two triangles have the same area AND the same perimeter, must they be congruent? If the answer is no, could you please provide me with a counterexample? Thanks a lot. I think the answer is no, but I have been unable to find a counterexample Doug Lewis Date: 6/26/96 at 14:57:3 From: Doctor Ceeks Subject: Re: Congruent Triangles Heron's formula gives: A^2 = s(s-a)(s-b)(s-c), where s is the semi-perimeter (a+b+c)/2. The question is: do there exist a,b,c and a',b',c' which satisfy the triangle inequalities for which a+b+c = a'+b'+c' and (s-a)(s-b)(s-c) = (s-a')(s-b')(s-c') where we define 2s = a+b+c? Because the set of a,b,c which satisfies the triangle inequality forms a 3-dimensional space, and because fixing the area and the perimeter eliminate a total of two degrees of freedom, one can expect the answer to be yes. And in fact, a little playing around gives: 5,12,13 and 9, (21-root(41))/2, (21+root(41))/2 as the sides of two triangles with the same area and perimeter, and in fact the area and perimeter are both 30. -Doctor Ceeks, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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