Find a point U in a square QRST such that QU=2, TU=1, and RU=3. Find the area of QTUR?

Question

without finding the length of the side of the square.

It is possible, for example, if the length of the side of the square was two, there would be two triangles with three sides and you could figure out the area. However, the length of the side is not necessarily an integer, and is not supposed to come into play in the problem.

Answer 1

This requires a circle of radius 3 with center R to be tangent to a circle of radius 1 with center T at point U and a circle of radius 2 with center Q to pass through point U. It also requires QRST to be a square.

This means that Q must lie on the diagonal RT whose length is 3+1 = 4. The point Q Has a y coordinate of 3cos45
= 3sqrt(2)/2 and an x coordinate of 2sqrt(2)- 3sqrt(2)/2 =
-sqrt(2) which is impossible.

Thus there is no solution.