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I have a problem that I encountered on a math competition, but I don't know how to solve it. Can someone help me?
A scalene trapezoid ABCD with bases AB and CD has a perimeter of 10 centimeters. All the side lengths are integer numbers. Find the height of this trapezoid in simplest radical form.
No, I don't need the answer, just the solution.
Cut out the center rectangle from the trapezoid and glue the two right triangles together to form triangle $ABC$. Notice that $ABC$ needs to have integer sides, cannot be equilateral, and that we've removed an even length from the perimeter, so $ABC$ has even perimeter less than $10$. There is only one such triangle.
If the side lengths are $a,b,c,d$, we know that $a+b+c+d=10$. Also, scalene means that the numbers $a,b,c,d$ are different. This determines $a,b,c,d$ up to permutation. If $a$ is the bottom and $c$ is the (smaller) top, then the height of the trapezoid is also the height of the triangle with side lengths $a-c,b,d$. A triangle with given side lengths is only possible if the longest side is shorter than the sum of the two other sides. You will notice that only one triangle is possible under these restrictions. To find its heiught, Heron's area formula might be helpful.
