SOLUTION: Prove that if the diagonals of a parallegram are equal it is a rectangle.

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Question 1022906: Prove that if the diagonals of a parallegram are equal it is a rectangle.
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Consider the parallelogram ABCD on the Cartesian plane, with coordinates A(0,0), B(a,0), C(c+a,d), and D(c,d). Without loss of generality, we can assume a>0 and c%3E=0. (We wish to show that c = 0.)
The two diagonals are AC and BD.
Now apply the distance formula to the endpoints of AC and BD:
abs%28AC%29%5E2+=+abs%28BD%29%5E2
==> %28c%2Ba%29%5E2+%2B+d%5E2+=+%28c-a%29%5E2%2Bd%5E2
==> c%5E2+%2B2ac%2Ba%5E2+%2Bd%5E2+=+c%5E2+-2ac%2Ba%5E2+%2B+d%5E2, after expansion;
==> 4ac = 0, after simplification.
Since a>0, we then have c = 0. (This implies that angle DAB and angle ABC are right angles.)
Therefore parallelogram ABCD is a rectangle.