The radius of the larger circle is $10$ cm.
Find the radius of the largest circle that will fit in the middle.
From my IGCSE math textbook.
I have tried to solve it but its too hard for me.
First I tried making a square by joining the smaller diameters.One diameter plus the distance from it to to centre=x. The answer is $\sqrt{2x^2}$ But how to find x, I don't know.
Can someone come up with a solution that's understandable and suitable for me? I am an IGCSE (Extended) student.
On
Denote the diameters $D_2 < D_1 < D_0$. We immediately find $$D_0 = 2D_1 + D_2.$$ Moreover, connecting the centers of the four smaller circles yields a square with side length $D_1$. Thus we can compute its diagonal by $$\sqrt{2D_1^2} = D_1 + D_2.$$ Solving for $D_1$ we get $$D_1 = \dfrac{1}{\sqrt{2}-1} D_2.$$ Plugging this into the first equation lets you compute the desired value $D_2$.
Consider the square formed by the centers of the four medium circles. Its side is the diameter of the medium circles.
The diameters of the large and small circles are the diagonal of the square, plus or minus the diameter of a medium circle.
Hence the small diameter
$$10\frac{\sqrt2-1}{\sqrt2+1}.$$