Problem 14 · Two squares in a circle
Coordinates with the centre on a side: the radius comes from a 3-4-5.
Integer answer, at most 4 digitsTwo squares with sides 3 cm and 4 cm are drawn inside a circle using 6 segments as shown in the figure, so that the centre of the circle lies on a side of the larger square. What is the radius of the circle?
Reasoned solution
Key idea: set up coordinates with the centre $O$ at the origin. Requiring the two extreme vertices to lie on the circle pins down the position of the squares, and the radius comes from a $3$-$4$-$5$ triangle.
The two squares share a vertex $J$ and have sides on the same horizontal line (that is why the figure is drawn with only $6$ segments: two pairs of sides are collinear). Put that line on the $x$-axis, with $O = (0,0)$ (the centre lies on the top side of the large square) and $J = (j, 0)$.
The small square occupies $[j-3,\, j] \times [0,3]$ (above the line) and the large one $[j,\, j+4] \times [-4,0]$ (below). The vertices touching the circle are the farthest ones: the top-left corner of the small square, $(j-3,\ 3)$, and the bottom-right corner of the large one, $(j+4,\ -4)$. Impose that they are equidistant from $O$:
The vertices end up at $(-4,\ 3)$ and $(3,\ -4)$, and the radius is